Add files using upload-large-folder tool

parse/train/1flmvXGGJaa/1flmvXGGJaa.md

@@ -0,0 +1,505 @@
+# NAS-BENCH-301 AND THE CASE FOR SURROGATE BENCHMARKS FOR NEURAL ARCHITECTURE SEARCH
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The most significant barrier to the advancement of Neural Architecture Search (NAS) is its demand for large computational resources, which hinders scientifically sound empirical evaluations. As a remedy, several tabular NAS benchmarks were proposed to simulate runs of NAS methods in seconds. However, all existing tabular NAS benchmarks are limited to extremely small architectural spaces since they rely on exhaustive evaluations of the space. This leads to unrealistic results that do not transfer to larger search spaces. To overcome this fundamental limitation, we propose NAS-Bench-301, the first surrogate NAS benchmark, using a search space containing $1 0 ^ { 1 8 }$ architectures, many orders of magnitude larger than any previous tabular NAS benchmark. After motivating the benefits of a surrogate benchmark over a tabular one, we fit various regression models on our dataset, which consists of ${ \sim } 6 0 \mathrm { k }$ architecture evaluations, and build surrogates via deep ensembles to also model uncertainty. We benchmark a wide range of NAS algorithms using NAS-Bench-301 and obtain comparable results to the true benchmark at a fraction of the real cost. Finally, we show how NAS-Bench-301 can be used to generate new scientific insights.
+
+# 1 INTRODUCTION
+
+Neural Architecture Search (NAS) promises to advance representation learning by automatically finding architectures that facilitate the learning of strong representations for a given dataset. NAS has already achieved state-of-the-art performance on many tasks (Real et al., 2019; Liu et al., 2019a; Saikia et al., 2019; Elsken et al., 2020) and to create resource-aware architectures (Tan et al., 2018; Elsken et al., 2019a; Cai et al., 2020). For a review, we refer to Elsken et al. (2019b).
+
+Despite many advancements in terms of both efficiency and performance, empirical evaluations in NAS are still problematic. Different NAS papers often use different training pipelines, different search spaces and different hyperparameters, do not evaluate other methods under comparable settings, and cannot afford enough runs for testing significance. This practice impedes assertions about the statistical significance of the reported results, recently brought into focus by several authors (Yang et al., 2019; Lindauer & Hutter, 2019; Shu et al., 2020; Yu et al., 2020).
+
+To circumvent these issues and enable scientifically sound evaluations in NAS, several tabular benchmarks (Ying et al., 2019; Zela et al., 2020b; Dong & Yang, 2020; Klyuchnikov et al., 2020) have been proposed recently (see also Appendix A.1 for more details). However, all these benchmarks rely on an exhaustive evaluation of all architectures in a search space, which limits them to unrealistically small search spaces (so far containing only between 6k and $4 2 3 \mathrm { k }$ architectures). This is a far shot from standard spaces used in the NAS literature, which contain more than $1 0 ^ { 1 8 }$ architectures (Zoph & Le, 2017; Liu et al., 2019b). This discrepancy can cause results gained on existing tabular NAS benchmarks to not generalize to realistic search spaces; e.g., promising anytime results of local search on existing tabular NAS benchmarks were shown to not transfer to realistic search spaces (White et al., 2020b). To address these problems, we make the following contributions:
+
+1. We present NAS-Bench-301, a surrogate NAS benchmark that is first to cover a realistically-sized search space (namely the cell-based search space of DARTS (Liu et al., 2019b)), containing more than $1 0 ^ { 1 8 }$ possible architectures. This is made possible by estimating their performance via a surrogate model, removing the constraint to exhaustively evaluate the entire search space.
+
+2. We empirically demonstrate that a surrogate fitted on a subset of architectures can in fact model the true performance of architectures better than a tabular benchmark (Section 2). 3. We analyze and release the NAS-Bench-301 training dataset consisting of ${ \sim } 6 0 \mathrm { k }$ fully trained and evaluated architectures, which will also be publicly available in the Open Graph Benchmark (Hu et al., 2020) (Section 3). 4. Using this dataset, we thoroughly evaluate a variety of regression models as surrogate candidates, showing that strong generalization performance is possible even in large spaces (Section 4). 5. We utilize NAS-Bench-301 as a benchmark for running various NAS optimizers and show that the resulting search trajectories closely resemble the ground truth trajectories. This enables sound simulations of thousands of GPU hours in a few seconds on a single CPU machine (Section 5). 6. We demonstrate that NAS-Bench-301 can help in generating new scientific insights by studying a previous hypothesis on the performance of local search in the DARTS search space (Section 6).
+
+To foster reproducibility, we open-source all our code and data in a public repo: https:// anonymous.4open.science/r/3f99ef91-c472-4394-b666-5d464e099aca/
+
+# 2 MOTIVATION – CAN WE DO BETTER THAN A TABULAR BENCHMARK?
+
+We start by motivating the use of surrogate benchmarks by exposing an issue of tabular benchmarks that has largely gone unnoticed. Tabular benchmarks are built around a costly, exhaustive evaluation of all possible architectures in a search space, and when an architecture’s performance is queried, the tabular benchmark simply returns the respective table entry. The issue with this process is that the stochasticity of mini-batch training is also reflected in the performance of an architecture $i$ , hence making it a random variable $Y _ { i }$ . Therefore, the table only contains results of a few draws $y _ { i } \sim Y _ { i }$ (existing NAS benchmarks feature up to 3 runs per architecture). Given the variance in these evaluations, a tabular benchmark acts as a simple estimator that assumes independent random variables, and thus estimates the performance of an architecture based only on previous evaluations of the same architecture. From a machine learning perspective, knowing that similar architectures tend to yield similar performance, and that the variance of individual evaluations can be high (both shown to be the case by Ying et al. (2019)), it is natural to assume that better estimators may exist. In the remainder of this section, we empirically verify this hypothesis and show that surrogate benchmarks can provide better performance estimates than tabular benchmarks based on less data.
+
+Setup We choose NAS-Bench-101 (Ying et al., 2019) as a tabular benchmark for our analysis and a Graph Isomorphism Network (GIN, $\mathrm { X u }$ et al. (2019a)) as our surrogate model.1 Each architecture $x _ { i }$ in NAS-Bench-101 contains 3 validation accuracies $y _ { i } ^ { 1 } , y _ { i } ^ { 2 } , y _ { i } ^ { 3 }$ from training $x _ { i }$ with 3 different seeds. We excluded all diverged models with less than $50 \%$ validation accuracy on any of the three evaluations in NAS-Bench-101. We split this dataset to train the GIN surrogate model on one of the seeds, e.g., $\mathcal { D } ^ { t r a i n } = \{ ( x _ { i } , y _ { i } ^ { 1 } ) \} _ { i }$ and evaluate on the other two, e.g., $\mathcal { D } ^ { t e s t } = \{ ( x _ { i } , \bar { y } _ { i } ^ { 2 3 } ) \} _ { i }$ , where $\bar { y } _ { i } ^ { 2 3 } = ( y _ { i } ^ { 2 } + y _ { i } ^ { 3 } ) / 2$ .
+
+<table><tr><td>Model</td><td colspan="3">Mean Absolute Error (MAE)</td></tr><tr><td></td><td>1,[2,3]</td><td>2,[1,3]</td><td>3,[1,2]</td></tr><tr><td>Tab.</td><td>4.534e-3</td><td>4.546e-3</td><td>4.539e-3</td></tr><tr><td>Surr.</td><td>3.446e-3</td><td>3.455e-3</td><td>3.441e-3</td></tr></table>
+
+Table 1: MAE between performance predicted by a tab./surr. benchmark fitted with one seed each, and the true performance of evaluations with the two other seeds. Test seeds in brackets.
+
+We emphasize that training a surrogate to model a search space is not a typical inductive regression task but rather a transductive one. By definition of the search space, the set of possible architectures is known ahead of time (although it may be very large), hence a surrogate model does not have to generalize to out-ofdistribution data if the training data covers the space well.
+
+Results We compute the mean absolute error MAE = Pi |yˆi−y¯23i | on $\mathcal { D } ^ { t r a i n } = \{ ( x _ { i } , y _ { i } ^ { 1 } ) \} _ { i }$ , where $\hat { y } _ { i }$ is predicted ofnd $n = | \mathcal { D } ^ { t e s t } |$ e model trained. Table 1 shows that the surrogate model yields a lower MAE than the tabular benchmark, i.e. MAE = Pi |y1i −y¯23i |n . We also report the mean squared error and Kendall tau correlation coefficient in Table 6 in the Appendix showing that the ranking between architectures is also predicted better by the surrogate.
+
+We repeat the experiment in a cross-validation fashion w.r.t to the seeds and conclude: In contrast to a single tabular entry, the surrogate model learns to smooth out the noise.2
+
+Next, we fit the GIN surrogate on subsets of $\mathcal { D } ^ { t r a i n }$ and plot how its performance scales with the amount of training data used in Figure 1. The surrogate model performs better than the tabular benchmark when the training set has more than $\sim 2 1 { , } 5 0 0$ architectures. Note that $\bar { \mathcal { D } } ^ { t e s t }$ remains the same as in the previous experiment, i.e., it includes all architectures in NAS-Bench-101. As a result, we conclude that: A surrogate model can yield strong predictive performance when only a subset of the search space is available as training data.
+
+These empirical findings suggest that we can create reliable surrogate benchmarks for much larger and more realistic NAS spaces, which are infeasible to be exhaustively evaluated as done by tabular benchmarks. In the remainder of the paper, we focus on creating such a benchmark.
+
+![](images/e1666a6d80f61bd1650dd826ad4be2627b9d4534dcf0cd1d0f2a53cdd3f74e56.jpg)  
+Figure 1: Number of architectures used for training the GIN surrogate model vs MAE on the NAS-Bench-101 dataset.
+
+# 3 THE NAS-BENCH-301 DATASET
+
+We now describe the NAS-Bench-301 dataset which consists of ${ \sim } 6 0 \mathrm { k }$ architectures and their performances on CIFAR-10 (Krizhevsky, 2009) sampled from the most popular NAS cell search space: the one from DARTS (Liu et al., 2019b). We use this dataset not only to fit surrogate models but also to gain new insights, such as which regions of the architecture space are being explored by different NAS methods, or what the characteristics of architectures are that work well.
+
+# 3.1 DATA COLLECTION
+
+<table><tr><td>NAS methods</td><td></td><td># eval</td></tr><tr><td></td><td>RS (Bergstra &amp; Bengio,2012)</td><td>23746</td></tr><tr><td rowspan="2">Evolution</td><td>DE (Awad et al.,2020)</td><td>7275</td></tr><tr><td>RE (Real et al.,2019)</td><td>4639</td></tr><tr><td>BO</td><td>TPE (Bergstra et al., 2011) BANANAS (White et al.,2019) COMBO (Oh et al., 2019)</td><td>6741 2243 745</td></tr><tr><td>One-Shot</td><td>DARTS (Liu et al., 2019b) PC-DARTS (Xu et al.,2020) DrNAS (Chen et al.,2020)</td><td>2053 1588 947</td></tr></table>
+
+Table 2: NAS methods used to cover the search space.
+
+Since the DARTS search space (detailed description in Appendix C.1) is far too large to be exhaustively evaluated, care has to be taken when sampling the architectures which will be used to train the surrogate models. Sampling should yield a good overall coverage of the architecture space while also providing a special focus on the well-performing regions that optimizers tend to exploit.
+
+Our principal methodology is inspired by Eggensperger et al. (2015), who collected unbiased data about hyperparameter spaces by random search, as well as biased and dense samples in high-performance regions by running hyperparameter optimizers. This is desirable for a surrogate benchmark since we are interested in evaluating NAS methods that exploit such good regions of the space. Table 2 lists the NAS methods we used to collect such samples and the respective number of samples. Additionally, we evaluated ${ \sim } 1 \mathrm { k }$ architectures in poorly-performing regions for better coverage and another ${ \sim } 1 0 \mathrm { k }$ for the analysis conducted on the dataset and surrogates. We refer to Appendices C.2 and C.3 for details on the data collection and the optimizers, respectively. We would like to point out that in hindsight adding training data of well-performing regions may be less important for a surrogate NAS benchmark than for a surrogate HPO benchmark, which we demonstrated in Appendix E.3. We argue that this is a result of HPO search spaces containing many configurations which yield disfunctional models, which is less common for architectures in many NAS search spaces, hence allowing random search to give us good coverage of the space.
+
+![](images/dacf59af9cb377fbef6adb2e4177ead28249bffd6bc9740a221694a717a20874.jpg)  
+Figure 2: t-SNE visualization of the sampled architectures.
+
+In Figure 2, we visualize the overall coverage of the search space as well as the similarity between sampled architectures using t-SNE (van der Maaten & Hinton, 2008). Besides showing a good overall coverage, some well-performing architectures in the search space form distinct clusters which are mostly located outside the main cloud of points. This clearly indicates that architectures with similar performance are close to each other in the architecture space. Additionally, we observe that different optimizers sample different types of architectures, see Figure 9 in the Appendix.
+
+# 3.2 PERFORMANCE STATISTICS
+
+Figure 3 shows the validation error on CIFAR-10 (Krizhevsky, 2009) of all sampled architectures in relation to the model parameters and training runtime. Generally, as expected, models with more parameters are more costly to train but achieve lower validation errors. We also find that different NAS methods yield quite different performance distributions (see Appendix C.4 for their individual performances). Validation and test errors are highly correlated with a Kendall tau rank correlation of $\tau = 0 . 8 5 2$ (Spearman rank corr. 0.969), minimizing the risk of overfitting on the validation error.
+
+Furthermore, we find that cells of all depths can reach a good performance, but shallow topologies are slightly favored in our setting (see Figure 10 in the Appendix). Also, a small number of parameter-free operations (e.g., skip connections) can benefit the performance but featuring many of these significantly deteriorates performance. For the full analysis, see Appendix C.5.
+
+Following standard practice in modern NAS papers (e.g., Liu et al. (2019b)), we employ various data augmentation techniques during training for more reliable estimates of an architecture’s performance. For a description of our full training pipeline, please see Appendix C.6.
+
+# 3.3 NOISE IN ARCHITECTURE EVALUATIONS
+
+As discussed in Section 2, the noise in architecture evaluations can be large enough for surrogate models to yield more realistic estimates of architecture performance than a tabular benchmark based on a single evaluation per architecture. To study the magnitude of this noise on NAS-Bench-301, we evaluated 500 architectures randomly sampled from our Differential Evolution (DE) (Awad et al., 2020) run with 5 different seeds each.3 We find a mean standard deviation of $1 . 6 \mathrm { { e } - 3 }$ for the final validation accuracy which is slightly less than the noise observed in NAS-Bench-101 (Ying et al., 2019); one possible reason for this could be a more robust training pipeline. Figure 12 in the Appendix shows that, while the noise tends to be lower for the best architectures, a correct ranking based on a single evaluation is still difficult. Finally, we compare the MAE when estimating the architecture performance from only one sample to the results from Table 1. Here, we also find a slightly lower MAE of $1 . 3 8 \mathrm { e } { - 3 }$ than for NAS-Bench-101.
+
+![](images/59c4329f48205ba53b8ee83482166dab9c8d5f1e3ba437c87fe58d5e4c371850.jpg)  
+Figure 3: Number of parameters against val. error with model training time as colorbar.
+
+# 4 FITTING SURROGATE MODELS ON THE NAS-BENCH-301 DATASET
+
+We now focus on creating a surrogate model. To that end, we evaluated a wide range of regression models on the NAS-Bench-301 dataset. In principle, any such model can give rise to a surrogate NAS benchmark, but models that fit the true performance better yield surrogate NAS benchmarks whose characteristics are more similar to the ones of the true benchmark. Therefore, we naturally strive for the best-fitting model. We emphasize that in this work we do not attempt to introduce a new regression model but rather build on the shoulders of the architecture performance prediction community.
+
+# 4.1 SURROGATE MODEL CANDIDATES
+
+Deep Graph Convolutional Neural Networks are frequently used as NAS predictors (Friede et al., 2019; Wen et al., 2019; Ning et al., 2020). In particular, we choose the GIN since several works have found it to perform well on many benchmark datasets (Errica et al., 2020; Hu et al., 2020; Dwivedi et al., 2020). We use the publicly available implementation from the Open Graph Benchmark (Hu et al., 2020) and refer to Appendix D.2 for further details.
+
+We compare the GIN to a variety of common regression models. We evaluate Random Forests (RF) and Support Vector Regression (SVR) using implementations from scikit-learn (Pedregosa et al., 2011). We also compare to the tree-based gradient boosting methods XGBoost (Chen & Guestrin, 2016). LGBoost (Ke et al., 2017) and NGBoost (Duan et al., 2020), recently used for predictorbased NAS (Luo et al., 2020). We comprehensively review architecture performance prediction in Appendix A.2.
+
+# 4.2 EVALUATING THE DATA FIT
+
+Similarly to Wen et al. (2019) and Baker et al. (2017), we assess the quality of the data fit via the coefficient of determination $( R ^ { 2 } )$ and the Kendall rank correlation coefficient $( \tau )$ . Since Kendall $\tau$ is sensitive to noisy evaluations that change the rank of an architecture, we follow the recent work by $\mathrm { Y u }$ et al. (2020) and use a sparse Kendall Tau (sKT), which ignores rank changes at $0 . 1 \%$ accuracy precision, by rounding the predicted validation accuracy prior to computing $\tau$ .
+
+All hyperparameters of the surrogate models were tuned using BOHB (Falkner et al., 2018) as a black-box optimizer; details on their respective hyperparameter search spaces are given in Table 7 in the appendix. We use train/val/test splits $( 0 . 8 / 0 . 1 / 0 . 1 )$ stratified across the NAS methods used for the data collection. This means that the ratio of architectures from a particular optimizer is constant across the splits, e.g. the test set contains $50 \%$ of its architectures from RS since RS was used to obtain $50 \%$ of the total architectures we trained and evaluated. We provide additional details on the preprocessing of the architectures for the surrogate models in Appendix D.1. As Table 3 shows, the three best-performing models are LGBoost, XGBoost and GIN; we therefore focus our analysis on these in the following.
+
+Table 3: Performance of different regression models fitted on the NB-301 dataset.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Test</td></tr><tr><td>R²</td><td>sKT</td></tr><tr><td>LGBoost</td><td>0.892</td><td>0.816</td></tr><tr><td>XGBoost</td><td>0.832</td><td>0.817</td></tr><tr><td>GIN</td><td>0.832</td><td>0.778</td></tr><tr><td>NGBoost</td><td>0.810</td><td>0.759</td></tr><tr><td>μ-SVR</td><td>0.709</td><td>0.677</td></tr><tr><td>MLP(Path enc.)</td><td>0.704</td><td>0.697</td></tr><tr><td>RF</td><td>0.679</td><td>0.683</td></tr><tr><td>E-SVR</td><td>0.675</td><td></td></tr><tr><td></td><td></td><td>0.660</td></tr></table>
+
+In addition to evaluating the data fit on our data splits, we investigate the impact of parameterfree operations and the cell topology in Appendices D.6 and D.7, respectively. We find that all of LGBoost, XGBoost and GIN accurately predict the drop in performance when increasingly replacing operations with parameter-free operations in a normal cell.
+
+Table 4: Leave One-Optimizer-Out performance of the best surrogate models.   
+
+<table><tr><td></td><td>Model</td><td>NoRE</td><td>NoDE</td><td>No COMBO</td><td>No TPE</td><td>No BANANAS</td><td>No DARTS</td><td>No PC-DARTS</td><td>No DrNAS</td><td>No GDAS</td></tr><tr><td>R²</td><td>LGB</td><td>0.917</td><td>0.892</td><td>0.919</td><td>0.857</td><td>0.909</td><td>-0.093</td><td>0.826</td><td>0.699</td><td>0.429</td></tr><tr><td></td><td>XGB</td><td>0.907</td><td>0.888</td><td>0.876</td><td>0.842</td><td>0.911</td><td>-0.151</td><td>0.817</td><td>0.631</td><td>0.672</td></tr><tr><td></td><td>GIN</td><td>0.856</td><td>0.864</td><td>0.775</td><td>0.789</td><td>0.881</td><td>0.115</td><td>0.661</td><td>0.790</td><td>0.572</td></tr><tr><td></td><td>LGB</td><td>0.834</td><td>0.782</td><td>0.833</td><td>0.770</td><td>0.592</td><td>0.780</td><td>0.721</td><td>0.694</td><td>0.595</td></tr><tr><td>sKT</td><td>XGB</td><td>0.831</td><td>0.780</td><td>0.817</td><td>0.762</td><td>0.596</td><td>0.775</td><td>0.710</td><td>0.709</td><td>0.638</td></tr><tr><td></td><td>GIN</td><td>0.798</td><td>0.757</td><td>0.737</td><td>0.718</td><td>0.567</td><td>0.765</td><td>0.645</td><td>0.706</td><td>0.607</td></tr></table>
+
+# 4.3 LEAVE ONE-OPTIMIZER-OUT ANALYSIS
+
+Since the aim of NAS-Bench-301 is to allow efficient benchmarking of novel NAS algorithms, it is necessary to ensure that the surrogate model can deliver accurate performance estimation on data from trajectories by unseen NAS methods. Similarly to Eggensperger et al. (2015), we therefore perform a form of cross-validation on the optimizers we used for data collection, i.e. we leave out all data collected by one of the NAS methods entirely during training (using a stratified $0 . 9 / 0 . 1$ train/val split over the other NAS methods). Then, we predict the unseen results from the left-out NAS method to evaluate how well the models extrapolate to the region covered by the ’unseen’ method. We refer to this as the leave-one-optimizer-out (LOOO) setting.
+
+Results The results in Table 4 show that the rank correlation between the predicted and observed validation accuracy remains high even when a well-performing optimizer such as RE is left out. Predicting BANANAS in the LOOO fashion yields a lower rank correlation, because it focuses on well-performing architectures that are harder to rank; however, the high $R ^ { 2 }$ shows that the fit is still good.
+
+Conversely, leaving out DARTS causes a low $R ^ { 2 }$ but still high sKT; this is due to architectures with many skip connections in the DARTS data that are overpredicted (further discussed in Section 5.2). For full details, Figure 16 in the appendix provides scatter plots of the predicted vs. true performance for each NAS method.
+
+# 4.4 NOISE MODELLING
+
+Ensemble methods are commonly used to improve predictive performance (Dietterich, 2000). Moreover, ensembles of deep neural networks, so-called deep ensembles, have been proposed as a simple way to obtain predictive uncertainty (Lakshminarayanan et al., 2017). We therefore create an ensemble of 10 base learners for each of our three best performing models (GIN, XGB, LGB) using a 10-fold cross-validation for our train and validation split, as well as different initializations. We use the architectures with multiple evaluations (see Section 3.3) to mirror the analysis in the motivation in Section 2. We train using only one evaluation per architecture (i.e., seed 1) and take the mean accuracy of the remaining ones as groundtruth (i.e., seeds 2-5). We then compare against a tabular model with just one evaluation (seed 1).
+
+Table 5 shows that the GIN and LGB surrogate models yield estimates closer to groundtruth than the table lookup based on one evaluation. This confirms our main finding from Section 2, but this time on a much larger search space. We also compare the predictive distribution of our ensembles to the groundtruth. To that end, we assume the noise in the architecture performance to be normally distributed and compute the Kullback–Leibler (KL) divergence between the groundtruth accuracy distribution and
+
+<table><tr><td>Model</td><td>MAE 1,[2,3,4,5]</td><td>Mean σ</td><td>KL div.</td></tr><tr><td>Tabular</td><td>1.38e-3</td><td>undef.</td><td>undef.</td></tr><tr><td>GIN</td><td>1.13e-3</td><td>0.6e-3</td><td>16.4</td></tr><tr><td>LGB</td><td>1.33e-3</td><td>0.3e-3</td><td>68.9</td></tr><tr><td>XGB</td><td>1.51e-3</td><td>0.3e-3</td><td>134.4</td></tr></table>
+
+Table 5: Metrics for the selected surrogate models on 500 architectures that were evaluated 5 times.
+
+predicted distribution. We find the GIN ensemble to quite clearly provide the best estimate.
+
+To allow evaluations of multi-objective NAS methods, and to allow using “simulated wallclock time” on the $\mathbf { X }$ axis of plots, we also predict the runtime of architecture evaluations. For this, we train an LGB model with the runtime as targets (see Appendix D.4 for details). Runtime prediction is less challenging than performance prediction, resulting in an excellent fit of our LGB runtime model on the test set (sKT: 0.936, $R ^ { 2 } \colon 0 . 9 8 7 $ ). Other metrics of architectures, such as the number of parameters and multiply-adds, do not require a surrogate model but can be queried exactly.
+
+# 5 NAS-BENCH-301 AS A SURROGATE NAS BENCHMARK
+
+Having assessed the ability of the surrogate models to model the search space, we now use NAS-Bench-301 to benchmark various NAS algorithms.
+
+![](images/a1dad8c2b8f2c23812425230a0b6d03f2d7005459a7f365517db0c5488c89224.jpg)  
+Figure 4: Anytime performance of different optimizers on the real benchmark (left) and the surrogate benchmark (GIN (middle) and XGB (right)) when training ensembles on data collected from all optimizers. Trajectories on the surrogate benchmark are averaged over 5 optimizer runs.
+
+![](images/fd3fe23d6de88edac422b9abeb5b9c96665e771d321854421ca846b1b5368f01.jpg)  
+Figure 5: Anytime performance of blackbox optimizers, comparing performance achieved on the real benchmark and on surrogate benchmarks built with GIN and XGB in an LOOO fashion.
+
+# 5.1 BLACKBOX OPTIMIZERS
+
+We first compare the trajectories on the true benchmark and on the surrogate benchmark for blackbox optimizers when training the surrogate on all data. For the true benchmark, we show the trajectories contained in our dataset (based on a single run, since we could not afford repetitions due to the extreme compute requirements of 115 GPU days for a single run). For the evaluations on the surrogate, on the other hand, we can trivially afford to perform multiple runs. For the surrogate trajectories, we use an identical initialization for the optimizers (e.g., initial population for RE) but evaluations of the surrogate benchmark are done by sampling from the surrogate model’s predictive distribution for the architecture at hand, leading to different trajectories.
+
+Results (all data) As Figure 4 shows, both the XGB and the GIN surrogate capture behaviors present on the true benchmark. For instance, the strong improvements of BANANAS and RE are also present on the surrogate benchmark at the correct time. In general, the ranking of the optimizers towards convergence is accurately reflected on the surrogate benchmark. Also, the initial random exploration of algorithms like TPE, RE and DE is captured as the large initial variation in performance indicates. Notably, the XGB surrogate ensemble exhibits a high variation in well-performing regions as well and seems to slightly underestimate the error of the best architectures. The GIN surrogate, on the other hand, shows less variance in these regions but slightly overpredicts for the best architectures.
+
+Note, that due to the size of the search space, random search stagnates and cannot identify one of the best architectures even after tens of thousands of evaluations, with BANANAS finding better architectures orders of magnitude faster. This stands in contrast to previous NAS benchmarks. For instance, NAS-Bench-201 (Dong & Yang, 2020) only contains 6466 unique architectures in total, causing the median of random search runs to find the best architecture after only 3233 evaluations.
+
+To simulate benchmarking of novel NAS methods, we expand on the leave-one-optimizer-out analysis (LOOO) from Section 4.3 and assess each optimizer with surrogate benchmarks based on data excluding that gathered by said optimizer. We again compare the trajectories obtained from 5 runs on the surrogate benchmark to the groundtruth.
+
+![](images/e38b57c16a82cfe4afff288ea19b94876fdd025e606d3c1f48d092ce28802614.jpg)  
+Figure 6: Anytime performance of one-shot optimizers, comparing performance achieved on the real benchmark and on surrogate benchmarks built with GIN and XGB in a LOOO fashion.
+
+Results (LOOO) Figure 5 shows the trajectories in the leave-one-optimizer-out setting. The XGB and GIN surrogates again capture the general behavior of different optimizers well, illustrating that characteristics of new optimization algorithms can be captured with the surrogate benchmark. Leaving out DE appears to be a bigger problem for XGB than GIN, pointing to advantages of the smooth embedding learned by the GIN compared to gradient-boosting.
+
+In an additional experiment, we found that surrogates built on only well-performing architectures $9 2 \%$ and above) yielded poor extrapolation to worse architectures, but that surrogate benchmarks based on them still yielded realistic trajectories. We attribute this to NAS optimizers’ focus on good architectures. For details, see Appendix E.2. We also investigate whether it is possible to create benchmarks only on random architectures in Appendix E.3, and find that we can indeed obtain realistic trajectories but lose some predictive performance in the well-performing regions. Nevertheless, such benchmarks have the advantage of not possibly favouring any NAS optimizer used for the generation of training data, and we thus recommend their release in addition to the benchmarks based on the full training data.
+
+# 5.2 ONE-SHOT OPTIMIZERS
+
+NAS-Bench-301 can also be used to monitor the behavior of one-shot NAS optimizers throughout their search phase, by querying the surrogate model with the currently most promising discrete architecture. This can be extremely useful in many scenarios since uncorrelated proxy and true objectives can lead to potential failure modes, e.g., to a case where the found architectures contain only skip connections in the normal cell (Zela et al., 2020a;b; Dong & Yang, 2020) (we study such a failure case in Appendix E.1 to ensure robustness of the surrogates in said case). We demonstrate this use case in a similar LOOO analysis as for the black-box optimizers, using evaluations of the discrete architectures from each search epoch of multiple runs of DARTS, PC-DARTS and GDAS as ground-truth. Figure 6 shows that the surrogate trajectories closely resemble the true trajectories.
+
+# 6 USING NAS-BENCH-301 TO DRIVE NAS RESEARCH
+
+We finally use our new benchmark to perform a case study that demonstrates how NAS-Bench301 can drive NAS research. Coming up with research hypotheses and drawing conclusions when prototyping or evaluating NAS algorithms on less realistic benchmarks is difficult, particularly when these evaluations require high computational budgets. NAS-Bench-301 alleviates this dilemma via its cheap and reliable estimates.
+
+To showcase such a scenario, we evaluate Local Search4 (LS) on our surrogate benchmark and the actual DARTS benchmark. White et al. (2020b) concluded that LS does not perform well on such a large space by running it for 11.8 GPU days $\approx 1 0 ^ { 6 }$ seconds), and we are able to reproduce the same results via NAS-Bench-301 in a few seconds (see Fig 7). While White et al. (2020b) could not afford longer runs (nor repeats), on NAS-Bench-301 this is trivial. Doing so suggests that LS shows qualitatively different behavior when run for an order of magnitude longer, transitioning from being the worst method to being one of the best. We verified this suggestion by running LS for longer on the actual DARTS benchmark (also see Fig 7). This allows us to revise the initial conclusion of White et al. (2020b) to: LS is also state-of-the-art for the DARTS search space, but only when given enough time.
+
+This case study shows how NAS-Bench-301 was already used to cheaply obtain hints on a research hypothesis that lead to correcting a previous finding that only held for short runtimes. We look forward to additional uses along such lines.
+
+# 7 CONCLUSIONS & GUIDELINES FOR USING NAS-BENCH-301
+
+![](images/824e3e70fe250cb9bbbfc54bc0f150b5be0523ee5ef05af4a8999e6818d42372.jpg)  
+Figure 7: Case study results for Local Search. GT is the ground truth, GIN and XGB are results on NAS-Bench-301.
+
+We proposed NAS-Bench-301, the first surrogate NAS benchmark and first to cover a realistic search space which is orders of magnitude larger than all previous tabular NAS benchmarks. After motivating the benefits of a surrogate benchmark over a tabular one, we described the strategy used to collect the data which we used to fit our selected surrogate models and evaluated their predictive performance. Lastly, we demonstrated that our surrogate benchmark can accurately simulate real anytime performance trajectories of various NAS methods at a fraction of the true cost and can lead to new scientific findings. We hope that NAS-Bench-301 will allow the NAS practitioner to quickly prototype and benchmark NAS algorithms on the currently most used search space, without requiring large computational resources. We also argue that NAS-Bench-301 could also be used to monitor one-shot optimizers during their search phase, to detect failure cases early on. Finally, the ideas and methods discussed in our work trivially transfer to other search spaces or datasets, allowing for the design of many interesting surrogate benchmarks in the future.
+
+Finally, we want to mention the risk that prior knowledge about the surrogate model in NAS-Bench-301 could lead to the design of algorithms that may overfit to the surrogate benchmark. To this end, we recommend the following best practices to ensure a safe and fair benchmarking of NAS methods on NAS-Bench-301 and future surrogate benchmarks:
+
+• The surrogate model should be treated as a black-box function, hence only be used for performance prediction and not exploited to extract, e.g., gradient information.   
+• We discourage benchmarking methods that internally use the same model as the surrogate model picked in NAS-Bench-301 (e.g. GNN-based Bayesian optimization should not only be benchmarked using the GIN surrogate benchmark).   
+• We encourage running experiments on versions of NAS-Bench-301 (and other, future NAS surrogate benchmarks) that are based on (1) all available training architectures and (2) only architectures collected with uninformed methods, such as random search or space-filling designs. As shown in Appendix E.3, (1) yields better predictive models, but (2) avoids any potential bias (in the sense of making more accurate predictions for architectures explored by a particular type of NAS optimizer) and can still yield strong benchmarks.   
+• In order to ensure comparability of results in different published papers, we ask users to state the benchmark’s version number. We will continuously collect more training data and further improve the surrogate model predictions. So far, we release NB301-XGB-v1.0, NB301-GINv1.0, NB301-XGB-rand-v1.0, and NB301-GIN-rand-v1.0.
+
+Due to the flexibility of surrogate NAS benchmarks to cover arbitrary search spaces, we expect NAS-Bench-301 to be the first of many such benchmarks. We collect best practices for the creation of new surrogate benchmarks in Appendix F. Having access to a variety of benchmarks is essential to the development and evaluation of new NAS methods. We therefore encourage the community to expand the scope of current NAS benchmarks to different search spaces, datasets, and problem domains utilizing surrogate benchmarks to cover large spaces.
+
+Acknowledgements We thank the anonymous reviewers for suggesting very insightful experiments, in particular the experiments for NAS benchmarks based only on random architectures.
+
+# REFERENCES
+
+N. Awad, N. Mallik, and F. Hutter. Differential evolution for neural architecture search (dehb). In International Conference on Machine Learning (ICLR) Neural Architecture Search (NAS) Workshop, 2020.
+
+B. Baker, O. Gupta, R. Raskar, and N. Naik. Accelerating Neural Architecture Search using Performance Prediction. In NeurIPS Workshop on Meta-Learning, 2017.
+
+R. Baptista and M. Poloczek. Bayesian optimization of combinatorial structures. In International Conference on Machine Learning (ICLR), pp. 462–471, 2018.
+
+J. Bergstra and Y. Bengio. Random search for hyper-parameter optimization. Journal of Machine Learning Research, 13:281–305, 2012.
+
+J. Bergstra, R. Bardenet, Y. Bengio, and B. Kegl. Algorithms for hyper-parameter optimization. In ´ J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K. Weinberger (eds.), Proceedings of the 25th International Conference on Advances in Neural Information Processing Systems (NIPS’11), pp. 2546–2554, 2011.
+
+H. Cai, C. Gan, T. Wang, Z. Zhang, and S. Han. Once-for-all: Train one network and specialize it for efficient deployment. In International Conference on Learning Representations (ICLR), 2020.
+
+T. Chen and C. Guestrin. Xgboost: A scalable tree boosting system. In Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining, pp. 785–794, 2016.
+
+X. Chen, R. Wang, M. Cheng, X. Tang, and C. Hsieh. Drnas: Dirichlet neural architecture search. arXiv preprint arXiv:2006.10355, 2020.
+
+P. Chrabaszcz, I. Loshchilov, and F. Hutter. A downsampled variant of imagenet as an alternative to the cifar datasets. arXiv preprint arXiv:1707.08819, 2017.
+
+B. Deng, J. Yan, and D. Lin. Peephole: Predicting network performance before training. arXiv preprint arXiv:1712.03351, 2017.
+
+T. DeVries and G. W. Taylor. Improved regularization of convolutional neural networks with cutout. arXiv preprint arXiv:1708.04552, 2017.
+
+T. Dietterich. Ensemble Methods in Machine Learning, volume 1857 of Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2000.
+
+T. Domhan, J. T. Springenberg, and F. Hutter. Speeding up automatic hyperparameter optimization of deep neural networks by extrapolation of learning curves. In Q. Yang and M. Wooldridge (eds.), Proceedings of the 25th International Joint Conference on Artificial Intelligence (IJCAI’15), pp. 3460–3468, 2015.
+
+X. Dong and Y. Yang. Searching for a robust neural architecture in four gpu hours. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1761–1770, 2019.
+
+X. Dong and Y. Yang. Nas-bench-102: Extending the scope of reproducible neural architecture search. In International Conference on Learning Representations (ICLR), 2020.
+
+T. Duan, A. Avati, D. Ding, Khanh K. Thai, S. Basu, A. Ng, and A. Schuler. Ngboost: Natural gradient boosting for probabilistic prediction. In Proceedings of Machine Learning and Systems 2020, pp. 6138–6148, 2020.
+
+V. P. Dwivedi, C. K. Joshi, T. Laurent, Y. Bengio, and X. Bresson. Benchmarking graph neural networks. arXiv preprint arXiv:2003.00982, 2020.
+
+K. Eggensperger, F. Hutter, H.H. Hoos, and K. Leyton-Brown. Efficient benchmarking of hyperparameter optimizers via surrogates. In B. Bonet and S. Koenig (eds.), Proceedings of the Twentynineth National Conference on Artificial Intelligence (AAAI’15), pp. 1114–1120. AAAI Press, 2015.
+
+T. Elsken, J. H. Metzen, and F. Hutter. Efficient multi-objective neural architecture search via lamarckian evolution. In International Conference on Learning Representations, 2019a.   
+T. Elsken, J. H. Metzen, and F. Hutter. Neural architecture search: A survey. Journal of Machine Learning Research, 20(55):1–21, 2019b.   
+T. Elsken, B. Staffler, J. Metzen, and F. Hutter. Meta-learning of neural architectures for fewshot learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12365–12375, 2020.   
+F. Errica, M. Podda, D. Bacciu, and A. Micheli. A fair comparison of graph neural networks for graph classification. In International Conference on Learning Representations, 2020.   
+S Falkner, A Klein, and F. Hutter. BOHB: Robust and efficient hyperparameter optimization at scale. In Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 1437–1446, 2018.   
+M. Fey and J. E. Lenssen. Fast graph representation learning with PyTorch Geometric. In ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019.   
+D. Friede, J. Lukasik, H. Stuckenschmidt, and M. Keuper. A variational-sequential graph autoencoder for neural architecture performance prediction. ArXiv, abs/1912.05317, 2019.   
+D. Golovin and A. Krause. Adaptive submodularity: Theory and applications in active learning and stochastic optimization. Journal of Artificial Intelligence Research, 42:427–486, 2011.   
+M. Gori, G. Monfardini, and F. Scarselli. A new model for learning in graph domains. In Proceedings. 2005 IEEE International Joint Conference on Neural Networks, 2005., volume 2, pp. 729–734. IEEE, 2005.   
+K. He, X. Zhang, S. Ren, and J. Sun. Deep Residual Learning for Image Recognition. In Computer Vision and Pattern Recognition (CVPR), 2016.   
+W. Hu, M. Fey, M. Zitnik, Y. Dong, H. Ren, B. Liu, M. Catasta, and J. Leskovec. Open graph benchmark: Datasets for machine learning on graphs. arXiv preprint arXiv:2005.00687, 2020.   
+J. Hwang, S. Lay, and A. Lippman. Nonparametric multivariate density estimation: a comparative study. IEEE Transactions on Signal Processing, 42(10):2795–2810, 1994.   
+R. Istrate, F. Scheidegger, G. Mariani, D. Nikolopoulos, C. Bekas, and A. C. I. Malossi. Tapas: Train-less accuracy predictor for architecture search. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 3927–3934, 2019.   
+G. Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, Q. Ye, and T. Liu. Lightgbm: A highly efficient gradient boosting decision tree. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems 30, pp. 3146–3154. Curran Associates, Inc., 2017.   
+T. N. Kipf and M. Welling. Semi-supervised classification with graph convolutional networks. In International Conference on Learning Representations (ICLR), 2017.   
+A. Klein, S. Falkner, J. T. Springenberg, and F. Hutter. Learning curve prediction with Bayesian neural networks. In Proceedings of the International Conference on Learning Representations (ICLR’17). CBLS, 2017.   
+N. Klyuchnikov, I. Trofimov, E. Artemova, M. Salnikov, M. Fedorov, and E. Burnaev. Nas-benchnlp: Neural architecture search benchmark for natural language processing. arXiv preprint arXiv:2006.07116, 2020.   
+A. Krizhevsky. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.   
+B. Lakshminarayanan, A. Pritzel, and C. Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. In Advances in neural information processing systems, pp. 6402– 6413, 2017.   
+G. Larsson, M. Maire, and G. Shakhnarovich. Fractalnet: Ultra-deep neural networks without residuals. In International Conference on Learning Representations (ICLR), 2017.   
+M. Lindauer and F. Hutter. Best practices for scientific research on neural architecture search. arXiv preprint arXiv:1909.02453, 2019.   
+M. Lindauer, K. Eggensperger, M. Feurer, A. Biedenkapp, J. Marben, P. Muller, and F. Hut- ¨ ter. Boah: A tool suite for multi-fidelity bayesian optimization & analysis of hyperparameters. arXiv:1908.06756 [cs.LG], 2019.   
+C. Liu, L. Chen, F. Schroff, H. Adam, W. Hua, A. Yuille, and L. Fei-Fei. Auto-deeplab: Hierarchical neural architecture search for semantic image segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 82–92, 2019a.   
+H. Liu, K. Simonyan, and Y. Yang. DARTS: Differentiable architecture search. In International Conference on Learning Representations, 2019b.   
+I. Loshchilov and F. Hutter. Sgdr: Stochastic gradient descent with warm restarts. In International Conference on Learning Representations (ICLR) 2017 Conference Track, April 2017.   
+R. Luo, X. Tan, R. Wang, T. Qin, E. Chen, and T. Liu. Neural architecture search with gbdt. arXiv preprint arXiv:2007.04785, 2020.   
+M. D. McKay, R. J. Beckman, and W. J. Conover. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 42(1): 55–61, 2000.   
+X. Ning, Y. Zheng, T. Zhao, Y. Wang, and H. Yang. A generic graph-based neural architecture encoding scheme for predictor-based nas. arXiv preprint arXiv:2004.01899, 2020.   
+C. Oh, J. Tomczak, E. Gavves, and M. Welling. Combinatorial bayesian optimization using the graph cartesian product. In Advances in Neural Information Processing Systems, pp. 2910–2920, 2019.   
+F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, et al. Scikit-learn: Machine learning in python. Journal of machine learning research, 12(Oct):2825–2830, 2011.   
+K. Price, R. M. Storn, and J. A. Lampinen. Differential evolution: a practical approach to global optimization. Springer Science & Business Media, 2006.   
+E. Real, A. Aggarwal, Y. Huang, and Q. V. Le. Regularized evolution for image classifier architecture search. In Proceedings of the aaai conference on artificial intelligence, volume 33, pp. 4780–4789, 2019.   
+T. Saikia, Y. Marrakchi, A. Zela, F. Hutter, and T. Brox. Autodispnet: Improving disparity estimation with automl. In IEEE International Conference on Computer Vision (ICCV), 2019.   
+H. Shi, R. Pi, H. Xu, Z. Li, J. T. Kwok, and T. Zhang. Multi-objective neural architecture search via predictive network performance optimization. arXiv preprint arXiv:1911.09336, 2019.   
+Y. Shu, W. Wang, and S. Cai. Understanding architectures learnt by cell-based neural architecture search. In International Conference on Learning Representations (ICLR), 2020.   
+I. M. Sobol’. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 7(4):784–802, 1967.   
+C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In Computer Vision and Pattern Recognition (CVPR), 2015.   
+M. Tan, B. Chen, R. Pang, V. Vasudevan, and Q. V. Le. Mnasnet: Platform-aware neural architecture search for mobile. 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2815–2823, 2018.   
+Y. Tang, Y. Wang, Y. Xu, H. Chen, B. Shi, C. Xu, C. Xu, Q. Tian, and C. Xu. A semi-supervised assessor of neural architectures. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.   
+L. van der Maaten and G. Hinton. Visualizing data using t-sne. Journal of Machine Learning Research (JMLR), 9(Nov):2579–2605, 2008.   
+W. Wen, H. Liu, H. Li, Y. Chen, G. Bender, and P. Kindermans. Neural predictor for neural architecture search. arXiv preprint arXiv:1912.00848, 2019.   
+C. White, W. Neiswanger, and Y. Savani. Bananas: Bayesian optimization with neural architectures for neural architecture search. arXiv preprint arXiv:1910.11858, 2019.   
+C. White, W. Neiswanger, S. Nolen, and Y. Savani. A study on encodings for neural architecture search. arXiv preprint arXiv:2007.04965, 2020a.   
+C. White, S. Nolen, and Y. Savani. Local search is state of the art for nas benchmarks. arXiv preprint arXiv:2005.02960, 2020b.   
+Z. Wu, S. Pan, F. Chen, G. Long, C. Zhang, and P. S. Yu. A comprehensive survey on graph neural networks. arXiv preprint arXiv:1901.00596, 2019.   
+K. Xu, W. Hu, J. Leskovec, and S. Jegelka. How powerful are graph neural networks? In International Conference on Learning Representations, 2019a.   
+Y. Xu, Y. Wang, K. Han, H. Chen, Y. Tang, S. Jui, C. Xu, Q. Tian, and C. Xu. Rnas: Architecture ranking for powerful networks. arXiv preprint arXiv:1910.01523, 2019b.   
+Y. Xu, L. Xie, X. Zhang, X. Chen, G. Qi, Q. Tian, and H. Xiong. Pc-darts: Partial channel connections for memory-efficient architecture search. In International Conference on Learning Representations, 2020.   
+A. Yang, P. M. Esperanc¸a, and F. M. Carlucci. Nas evaluation is frustratingly hard. arXiv preprint arXiv:1912.12522, 2019.   
+C. Ying, A. Klein, E. Christiansen, E. Real, K. Murphy, and F. Hutter. NAS-bench-101: Towards reproducible neural architecture search. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.), Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pp. 7105–7114, 2019.   
+K. Yu, R. Ranftl, and M. Salzmann. How to train your super-net: An analysis of training heuristics in weight-sharing nas. arXiv preprint arXiv:2003.04276, 2020.   
+A. Zela, T. Elsken, T. Saikia, Y. Marrakchi, T. Brox, and F. Hutter. Understanding and robustifying differentiable architecture search. In International Conference on Learning Representations, 2020a.   
+A. Zela, J. Siems, and F. Hutter. Nas-bench-1shot1: Benchmarking and dissecting one-shot neural architecture search. In International Conference on Learning Representations, 2020b.   
+H. Zhang, M. Cisse, Y. N. Dauphin, and D. Lopez-Paz. mixup: Beyond empirical risk minimization. International Conference on Learning Representations, 2018.   
+J. Zhou, G. Cui, Z. Zhang, C. Yang, Z. Liu, L. Wang, C. Li, and M. Sun. Graph neural networks: A review of methods and applications. arXiv preprint arXiv:1812.08434, 2018.   
+B. Zoph and Q. V. Le. Neural architecture search with reinforcement learning. In International Conference on Learning Representations (ICLR) 2017 Conference Track, 2017.
+
+# A RELATED WORK
+
+# A.1 EXISTING NAS BENCHMARKS
+
+Benchmarks for NAS were introduced only recently with NAS-Bench-101 (Ying et al., 2019) as the first among them. NAS-Bench-101 is a tabular benchmark consisting of ${ \sim } 4 2 3 \mathrm { k }$ unique architectures in a cell structured search space evaluated on CIFAR-10 (Krizhevsky, 2009). To restrict the number of architectures in the search space, the number of nodes and edges was given an upper bound and only three operations are considered. One result of this limitation is that One-Shot NAS methods can only be applied to subspaces of NAS-Bench-101 as demonstrated in NAS-Bench-1Shot1 (Zela et al., 2020b).
+
+NAS-Bench-201 (Dong & Yang, 2020), in contrast, uses a search space with a fixed number of nodes and edges, hence allowing for a straight-forward application of one-shot NAS methods. However, this limits the total number of unique architectures to as few as 6466. NAS-Bench-201 includes evaluations of all these architectures on three different datasets, namely CIFAR-10, CIFAR100 (Krizhevsky, 2009) and Downsampled Imagenet $1 6 \times 1 6$ (Chrabaszcz et al., 2017), allowing for transfer learning experiments.
+
+NAS-Bench-NLP (Klyuchnikov et al., 2020) was recently proposed as a tabular benchmark for NAS in the Natural Language Processing domain. The search space resembles NAS-Bench-101 as it limits the number of edges and nodes to constrain the search space size resulting in $1 4 \mathrm { k }$ evaluated architectures.
+
+# A.2 NEURAL NETWORK PERFORMANCE PREDICTION
+
+In the past, several works have attempted to predict the performance of neural networks by extrapolating learning curves (Domhan et al., 2015; Klein et al., 2017; Baker et al., 2017). A more recent line of work in performance prediction focuses more on feature encoding of neural architectures. Peephole (Deng et al., 2017) and TAPAS (Istrate et al., 2019) both use an LSTM to aggregate information about the operations in chain-structured architectures. On the other hand, BANANAS (White et al., 2019) introduces a path-based encoding of cells that automatically resolves the computational equivalence of architectures.
+
+Graph Neural Networks (GNNs) (Gori et al., 2005; Kipf & Welling, 2017; Zhou et al., 2018; Wu et al., 2019) with their capability of learning representations of graph-structured data appear to be a natural choice to learning embeddings of NN architectures. Shi et al. (2019) and Wen et al. (2019) trained a Graph Convolutional Network (GCN) on a subset of NAS-Bench-101 (Ying et al., 2019) showing its effectiveness in predicting the performance of unseen architectures. Moreover, Friede et al. (2019) propose a new variational-sequential graph autoencoder (VS-GAE) which utilizes a GNN encoder-decoder model in the space of architectures and generates valid graphs in the learned latent space.
+
+Several recent works further adapt the GNN message passing to embed architecture bias via extra weights to simulate the operations such as in GATES (Ning et al., 2020) or integrate additional information on the operations (e.g. flop count) (Xu et al., 2019b). Tang et al. (2020) chose to operate GNNs on relation graphs based on architecture embeddings in a metric learning setting, allowing to pose NAS performance prediction as a semi-supervised setting.
+
+# B TRAINING DETAILS FOR THE GIN IN THE MOTIVATION
+
+We set the GIN to have a hidden dimension of 64 with 4 hidden layers resulting in around ${ \sim } 4 0 \mathrm { k }$ parameters. We trained for 30 epochs with a batch size of 128. We chose the MSE loss function and add a logarithmic transformation to emphasize the data fit on well-performing architectures.
+
+Table 6: MSE and Kendall tau correlation between performance predicted by a tab./surr. benchmark fitted with one seed each, and the true performance of evaluations with the two other seeds (see Section 2). Test seeds in brackets.   
+
+<table><tr><td>Model</td><td colspan="3">Mean Squared Error (MSE)</td><td colspan="3">Kendall tau</td></tr><tr><td></td><td>1,[2,3]</td><td>2,[1,3]</td><td>3,[1,2]</td><td>1,[2,3]</td><td>2,[1,3]</td><td>3,[1,2]</td></tr><tr><td>Tab.</td><td>5.44e-5</td><td>5.43e-5</td><td>5.34e-5</td><td>0.83</td><td>0.83</td><td>0.83</td></tr><tr><td>Surr.</td><td>3.02e-5</td><td>3.07e-5</td><td>3.02e-5</td><td>0.87</td><td>0.87</td><td>0.87</td></tr></table>
+
+# C NAS-BENCH-301 DATASET
+
+# C.1 SEARCH SPACE
+
+We use the same architecture search space as in DARTS (Liu et al., 2019b). Specifically, the normal and reduction cell each consist of a DAG with 2 input nodes (receiving the output feature maps from the previous and previous-previous cell), 4 intermediate nodes (each adding element-wise feature maps from two previous nodes in the cell) and 1 output node (concatenating the outputs of all intermediate nodes). Input and intermediate nodes are connected by directed edges representing one of the following operations: Sep. conv $3 \times 3$ , Sep. conv $5 \times 5$ , Dil. conv $3 \times 3$ , Dil. conv $5 \times 5$ , Max pooling $3 \times 3$ , Avg. pooling $3 \times 3$ , Skip connection.
+
+# C.2 DATA COLLECTION
+
+To achieve good global coverage, we use random search to evaluate ${ \sim } 2 3 \mathrm { k }$ architectures. We note that space-filling designs such as quasi-random sequences, e.g. Sobol sequences (Sobol’, 1967), or Latin Hypercubes (McKay et al., 2000) and Adaptive Submodularity (Golovin & Krause, 2011) may also provide good initial coverage.
+
+Random search is supplemented by data which we collect from running a variety of optimizers, representing Bayesian Optimization (BO), evolutionary algorithms and One-Shot Optimizers. We used Tree-of-Parzen-Estimators (TPE) (Bergstra et al., 2011) as implemented by Falkner et al. (2018) as a baseline BO method. Since several recent works have proposed to apply BO over combinatorial spaces (Oh et al., 2019; Baptista & Poloczek, 2018) we also used COMBO (Oh et al., 2019). We included BANANAS (White et al., 2019) as our third BO method, which uses a neural network with a path-based encoding as a surrogate model and hence scales better with the number of function evaluations. As two representatives of evolutionary approaches to NAS, we chose Regularized Evolution (RE) (Real et al., 2019) as it is still one of the state-of-the art methods in discrete NAS and Differential Evolution (Price et al., 2006) as implemented by Awad et al. (2020). Accounting for the surge in interest in One-Shot NAS, our collected data collection also entails evaluation of architectures from search trajectories of DARTS (Liu et al., 2019b), GDAS (Dong & Yang, 2019), DrNAS (Chen et al., 2020) and PC-DARTS (Xu et al., 2020). For details on the architecture training details, we refer to Section C.6.
+
+For each architecture $a \in { \mathcal { A } }$ , the dataset contains the following metrics: train/validation/test accuracy, training time and number of model parameters.
+
+# C.3 DETAILS ON EACH OPTIMIZER
+
+In this section we provide the hyperparameters used for the evaluations of NAS optimizers for the collection of our dataset. Many of the optimizers require a specialized representation to function on an architecture space because most of them are general HPO optimizers. As recently shown by White et al. (2020a), this representation can be critical for the performance of a NAS optimizer. Whenever the representation used by the Optimizer did not act directly on the graph representation, such as in RE, we detail how we represented the architecture for the optimizer. All optimizers were set to optimize the validation error.
+
+BANANAS. We initialized BANANAS with 100 random architectures and modified the optimization of the surrogate model neural network, by adding early stopping based on a $90 \% / 1 0 \%$ train/validation split and lowering the number of ensemble models to be trained from 5 to 3. These changes to bananas avoided a computational bottleneck in the training of the neural network.
+
+COMBO. COMBO only attempts to maximize the acquisition function after the entire initial design (100 architectures) has completed. For workers which are done earlier, we sample a random architecture, hence increasing the initial design by the number of workers (30) we used for running the experiments. The search space considered in our work is larger than all search spaces evaluated in COMBO (Oh et al., 2019) and we regard not simply binary architectural choices, as we have to make choices about pairs of edges. Hence, we increased the number of initial samples for ascent acquisition function optimization from 20 to 30. Unfortunately, the optimization of the GP already became the bottleneck of the BO after around 600 function evaluations, leading to many workers waiting for new jobs to be assigned.
+
+Representation: In contrast to the COMBO’s original experimental setting, the DARTS search requires choices based on pairs of parents of intermediate nodes where the number of choices increase with the index of the intermediate nodes. The COMBO representation therefore consists of the graph cartesian product of the combinatorial choice graphs, increasing in size with each intermediate node. In addition, there exist 8 choices over the number of parameters for the operation in a cell.
+
+Differential Evolution. DE was started with a generation size of 100. As we used a parallelized implementation, the workers would have to wait for one generation plus its mutations to be completed for selection to start. We decided to keep the workers busy by training randomly sampled architectures in this case, as random architectures provide us good coverage of the space. However, different methods using asynchronous DE selection would also be possible. Note, that the DE implementation by Awad et al. (2020), performs boundary checks and resamples components of any individual that exceeds 1.0. We use the rand1 mutation operation which generally favors exploration over exploitation.
+
+Representation: DE uses a vector representation for each individual in the population. Categorical choices are scaled to lie within the unit interval [0, 1] and are rounded to the nearest category when converting back to the discrete representation in the implementation by Awad et al. (2020). Similarly to COMBO, we represent the increasing number of parent pair choices for the intermediate nodes by interpreting the respective entries to have an increasing number of sub-intervals in [0, 1].
+
+DARTS, GDAS, PC-DARTS and DrNAS. We collected the architectures found by all of the above one-shot optimizers with their default search hyperparameters. We performed multiple searches for each one-shot optimizer.
+
+RE. To allow for a good initial coverage before mutations start, we decided to randomly sample 3000 architectures as initial population. RE then proceeds with a sample size of 100 to extract well performing architectures from the population and mutates them. During mutations RE first decides whether to mutate the normal or reduction cell and then proceeds to perform either a parent change, an operation change or no mutation.
+
+![](images/8d8f9a0c20111a4e902b244568169d3e298cb1a9ffa8a2db6ccda80d1425cefc.jpg)  
+Figure 8: Empirical Cumulative Density Function (ECDF) plot comparing all optimizers in the dataset. Optimizers which cover good regions of the search space feature higher values in the low validation error region.
+
+TPE. For TPE we use the default settings as also used by BOHB. We use the Kernel-DensityEstimator surrogate model and build two models where the good configs are chosen as the top $15 \%$ . The acquisition function’s expected improvement is optimized by sampling 64 points.
+
+# C.4 OPTIMIZER PERFORMANCE
+
+The trajectories from the different NAS optimizers yield quite different performance distributions. This can be seen in Figure 8 which shows the ECDF of the validation errors of the architectures evaluated by each optimizer. As the computational budgets allocated to each optimizer vary widely, this data does not allow for a fair comparison between the optimizers. However, it is worth mentioning that the evaluations of BANANAS feature the best distribution of architecture performances, followed by PC-DARTS, DrNAS, DE, GDAS, and RE. TPE only evaluated marginally better architectures than RS, while COMBO and DARTS evaluated the worst architectures.
+
+We also perform a t-SNE analysis on the data collected by the different optimizers in Figure 9. We find that RE discovers well-performing architectures which form clusters distinct from the architectures found via RS. We observe that COMBO searched previously unexplored areas of the search space. BANANAS, which found some of the best architectures, explores clusters outside the main cluster. However, it heavily exploits regions at the cost of exploration. We argue that this is a result of the optimization of the acquisition function via random mutations based on the previously found iterates, rather than on new random architectures. DE is the only optimizer which finds well performing architectures in the center of the embedding space.
+
+# C.5 CELL TOPOLOGY, OPERATIONS AND NOISE
+
+In this section, we investigate the influence of the cell topology and the operations on the performance of the architectures in our setting. The discovered properties of the search space inform our choice of metrics for the evaluation of different surrogate models.
+
+First, we study how the validation error depends on the depth of architectures. Figure 10 visualizes the performance distribution of normal and reduction cells of different depth5 by approximating empirical distributions with a kernel density estimation used in violin plots (Hwang et al., 1994). We observe that the performance distributions are similar for the normal and reduction cells with the same cell depth. Although cells of all depths can reach high performances, shallower cells seem slightly favored. Note that these observations are subject to changes in the hyperparameter setting, e.g. training for more epochs may render deeper cells more competitive. The best-found architecture features a normal and reduction cell of depth 4. Color-coding the cell depth in our t-SNE projection also confirms that the t-SNE analysis captures the cell depth well as a structural property (c.f. Figure 13). It also reinforces that the search space is well-covered.
+
+We also show the distribution of normal and reduction cell depths of each optimizer in Figure 11 to get a sense for the diversity between the discovered architectures. We observe that DARTS and BANANAS generally find architectures with a shallow reduction cell and a deeper normal cell, while the reverse is true for RE. DE, TPE, COMBO and RS appear to find normal and reduction cells with similar cell depth.
+
+Aside from the cell topology, we can also use our dataset to study the influence of operations to the architecture performance. The DARTS search space contains operation choices without parameters such as Skip-Connection, Max Pooling $3 \times 3$ and Avg Pooling $3 \times 3$ . We visualize the influence of these parameter-free operations on the validation error in the normal and reduction cell in Figure 18a, respectively Figure 14. While pooling operations in the normal cell seem to have a negative impact on performance, a small number of skip connections improves the overall performance. This is somewhat expected, since the normal cell is dimension preserving and skip connections help training by improving gradient flow like in ResNets (He et al., 2016). In the reduction cell, the number of parameter-free operations has less effect as shown in Figure 14. In contrast to the normal cell where 2-3 skipconnections lead to generally better performance, the reduction cell shows no similar trend. For both cells, however, featuring many parameter-free operations significantly deteriorates performance. We therefore expect that a good surrogate also models this case as a poorly performing region.
+
+![](images/e20fd308f99a4d994eadcb307173665cf3c3056ae434bf25358a213bdcac73c6.jpg)  
+Figure 12: Standard deviation of the val. accuracy for multiple architecture evaluations.
+
+# C.6 TRAINING DETAILS
+
+Each architecture was evaluated on CIFAR-10 (Krizhevsky, 2009) using the standard 40k, 10k, 10k split for train, validation and test set. The networks were trained using SGD with momentum 0.9, initial learning rate of 0.025 and a cosine annealing schedule (Loshchilov & Hutter, 2017), annealing towards 10−8.
+
+We apply a variety of common data augmentation techniques which differs from previous NAS benchmarks where the training accuracy of many evaluated architectures reached $100 \%$ (Ying et al., 2019; Dong & Yang, 2020) indicating overfitting on the training set. We used CutOut (DeVries & Taylor, 2017) with cutout length 16 and MixUp (Zhang et al., 2018) with alpha 0.2. For regularization, we used an auxiliary tower (Szegedy et al., 2015) with a weight of 0.4 and DropPath (Larsson et al., 2017) with drop probability of 0.2. We trained each architecture for 100 epochs with a batch size of 96, using 32 initial channels and 8 cell layers. We chose these values to be close to the proxy model used by DARTS while also achieving good performance.
+
+![](images/bcc75f579509308689b4f53ccb48518748ed941e6b33ddecc8550e76f1f0fe3c.jpg)  
+Figure 9: Visualization of the exploration of different parts of the architectural t-SNE embedding space for all optimizers used for data collection. The architecture ranking by validation accuracy (lower is better) is global over the entire data collection of all optimizers.
+
+![](images/15e369a854ecd87d1cd9256b8e20183eaa7506248356df01ae7bdce089521bde.jpg)  
+Figure 10: Distribution of the validation error for different cell depth.
+
+![](images/0d032a7fd6b349a71c9b4dc64964da7a161ba4c017f043b7cf03f3e3864ac083.jpg)  
+Figure 11: Comparison between the normal and reduction cell depth for the architectures found by each optimizer.
+
+![](images/2898cb88ae3fc3862a9acb5d65bef3312ff9cfac3b29944499bf7ab4cbd35042.jpg)  
+Figure 13: t-SNE projection colored by the depth of the normal cell.
+
+![](images/4abf96fa94eb9d5ed112fcde09e7e5c6283aba467aff05d551ca07e92f920afb.jpg)  
+Figure 14: Distribution of validation error in dependence of the number of parameter-free operations in the reduction cell. Violin plots are cut off at the respective observed minimum and maximum value.
+
+# D SURROGATE MODEL ANALYSIS
+
+# D.1 PREPROCESSING OF THE GRAPH TOPOLOGY
+
+DGN preprocessing All DGN were implemented using PyTorch Geometric (Fey & Lenssen, 2019) which supports the aggregation of edge attributes. Hence, we can naturally represent the DARTS architecture cells, by assigning the embedded operations to the edges. The nodes are labeled as input, intermediate and output nodes. We represent the DARTS graph as shown in Figure 15, by connecting the output node of each cell type with the inputs of the other cell, allowing information from both cells to be aggregated per node during message passing. Note the self-loop on the output node of the normal cell, which we found necessary to get the best performance.
+
+Preprocessing for other surrogate models Since we make use of the framework implemented by BOHB (Falkner et al., 2018) to easily parallelize the architecture search algorithms across many compute nodes, we also represent our search space using ConfigSpace 6 (Lindauer et al., 2019). More precisely, we encode each pair of incoming edges for a cell as one choice of a categorical parameter. For instance, for node 4 in the normal cell, we add a parameter inputs node normal 4 with the choices of edge pairs 0 1,0 2,0 3,1 2,1 3,2 3. The edge operations are then implemented as categorical parameters for each edge and are only active if the corresponding edge was chosen. For instance, in the example above, if the incoming edge 0 is sampled, the parameter associated with the edge from node 0 to node 4 becomes activate and one operation is sampled. We provide the configuration space with our code. For all non-DGN based surrogate models, we use the vector representation of a configuration given by ConfigSpace as input to the model. This vector representation contains one value between 0 and 1 for each parameter in the configuration space.
+
+# D.2 DETAILS ON THE GIN
+
+The GIN implementation on the Open Graph Benchmark (OGB) (Hu et al., 2020) uses virtual nodes (additional nodes which are connected to all nodes in the graph) to boost performance as well as generalization and consistently achieves good performance on their public leaderboards. Other GNNs from Errica et al. (2020), such as DGCNN and DiffPool, performed worse in our initial experiments and are therefore not considered.
+
+Following recent work in Predictor-based NAS (Ning et al., 2020; Xu et al., 2019b), we use a per batch ranking loss because the ranking of an architecture is equally important to an accurate prediction of the validation accuracy in a NAS setting. We use the ranking loss formulation by GATES (Ning et al., 2020) which is a hinge pair-wise ranking loss with margin $\mathrm { { m } = 0 . 1 }$ .
+
+![](images/91b54aead24015d3711a9a7789cf9e4b61210279199da5dc09d016d28dbad341.jpg)  
+Figure 15: Architecture with inputs in green, intermediate nodes in blue and outputs in red.
+
+# D.3 DETAILS ON HPO
+
+All detailed table for the hyperparameter ranges for the HPO and the best values found by BOHB are listed in Table 7.
+
+# D.4 HPO FOR RUNTIME PREDICTION MODEL
+
+Our runtime prediction model is an LGB model trained on the runtimes of architecture evaluations of DE. This is because we partially evaluated the architectures utilizing different CPUs. Hence, we only choose to train on the evaluations carried out by the same optimizer on the same hardware to keep a consistent estimate of the runtime. DE is a good choice in this case because it both explored and exploited the architecture space well. The HPO space used for the LGB runtime model is the same used for the LGB surrogate model.
+
+# D.5 LEAVE ONE-OPTIMIZER-OUT ANALYSIS
+
+A detailed scatter plot of the predicted performance against the true performance for each optimizer and surrogate model in an LOOO analysis is provided in Figure 16 and Figure 17.
+
+# D.6 PARAMETER-FREE OPERATIONS
+
+Several works have found that methods based on DARTS (Liu et al., 2019b) are prone to finding sub-optimal architectures that contain many, or even only, parameter-free operations (max. pooling, avg. pooling or skip connections) and perform poorly (Zela et al., 2020a). We therefore evaluated the surrogate models on such architectures by replacing a random selection of operations in a cell with one type of parameter-free operations to match a certain ratio of parameter-free operations in a cell. This analysis is carried out over the test set of the surrogate models and hence contains architectures collected by all optimizers. For a more robust analysis, we repeated this experiment 4 times for each ratio of operations to replace.
+
+Results Figure 18 shows that both the GIN and the XGB model correctly predict that the accuracy drops with too many parameter-free operations, particularly for skip connections. The groundtruth of architectures with only parameter-free operations is displayed as scatter plot. Out of the two models,
+
+![](images/cf655209b256f8c7ddafa9bcc6c90f12b603465bca26be5fb1e7bce08a6a8a18.jpg)  
+Figure 16: Scatter plots of the predicted performance against the true performance of different surrogate models on the test set in a Leave-One-Optimizer-Out setting.
+
+![](images/41a684756b4c579f0d87328a855ddcef5eb1d83037d0fff2a0f034a7446942a4.jpg)  
+Figure 17: (continued) Scatter plots of the predicted performance against the true performance of different surrogate models on the test set in a Leave-One-Optimizer-Out setting.
+
+Table 7: Hyperparameters of the surrogate models and the default values found via HPO.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Hyperparameter</td><td rowspan="2">Range</td><td rowspan="2">Log-transform</td><td rowspan="2">Default Value</td></tr><tr><td></td></tr><tr><td rowspan="11">GIN</td><td>Hidden dim.</td><td>[16,256]</td><td>true</td><td>24</td></tr><tr><td>Num.Layers</td><td>[2,10]</td><td>false</td><td>8</td></tr><tr><td>Dropout Prob.</td><td>[0,1]</td><td>false</td><td>0.035</td></tr><tr><td>Learning rate</td><td>[1e-3,1e-2]</td><td>true</td><td>0.0777</td></tr><tr><td>Learning rate min.</td><td>const.</td><td>=</td><td>0.0</td></tr><tr><td>Batch size</td><td>const.</td><td></td><td>51</td></tr><tr><td>Undirected graph</td><td>[true, false]</td><td></td><td>false</td></tr><tr><td>Pairwise ranking loss</td><td>[true, false]</td><td></td><td>true</td></tr><tr><td>Self-Loops</td><td>[true, false]</td><td></td><td>false</td></tr><tr><td>Loss log transform</td><td>[true, false]</td><td></td><td>true</td></tr><tr><td>Node degree one-hot</td><td>const.</td><td></td><td>true</td></tr><tr><td rowspan="8">BANANAS</td><td>Num. Layers</td><td>[1,10]</td><td>true</td><td>17</td></tr><tr><td>Layer width</td><td>[16,256]</td><td>true</td><td>31</td></tr><tr><td>Dropout Prob.</td><td>const.</td><td></td><td>0.0</td></tr><tr><td>Learning rate</td><td>[le-3,le-1]</td><td>true</td><td>0.0021</td></tr><tr><td>Learning rate min.</td><td>const.</td><td></td><td>0.0</td></tr><tr><td>Batch size</td><td>[16,128]</td><td></td><td>122</td></tr><tr><td>Loss log transform</td><td>[true, false]</td><td></td><td>true</td></tr><tr><td>Pairwise ranking loss</td><td>[true, false]</td><td></td><td>false</td></tr><tr><td rowspan="11">XGBoost</td><td>Early Stopping</td><td>const.</td><td></td><td>100</td></tr><tr><td>Rounds Booster</td><td></td><td></td><td></td></tr><tr><td></td><td>const.</td><td></td><td>gbtree 13</td></tr><tr><td>Max.depth Min. child weight</td><td>[1,15]</td><td>false</td><td>39</td></tr><tr><td>Col.sample bylevel</td><td>[1,100]</td><td>true</td><td></td></tr><tr><td></td><td>[0.0,1.0]</td><td>false</td><td>0.6909</td></tr><tr><td>Col.sample bytree lambda</td><td>[0.0, 1.0]</td><td>false</td><td>0.2545</td></tr><tr><td>alpha</td><td>[0.001,1000] [0.001,1000]</td><td>true</td><td>31.3933 0.2417</td></tr><tr><td>Learning rate</td><td>[0.001,0.1]</td><td>true true</td><td>0.00824</td></tr><tr><td>Early stop. rounds</td><td></td><td></td><td></td></tr><tr><td></td><td>const.</td><td></td><td>100</td></tr><tr><td rowspan="11">LGBoost Random</td><td>Max.depth</td><td>[1,25]</td><td>false</td><td>18</td></tr><tr><td>Num. leaves</td><td>[10,100]</td><td>false</td><td>40</td></tr><tr><td>Max.bin</td><td>[100,400]</td><td>false</td><td>336</td></tr><tr><td>Feature Fraction</td><td>[0.1, 1.0]]</td><td>false</td><td>0.1532</td></tr><tr><td>Min. child weight</td><td>[0.001,10]</td><td>true</td><td>0.5822</td></tr><tr><td>Lambda L1</td><td>[0.001,1000]</td><td>true</td><td>0.0115</td></tr><tr><td>Lambda L2</td><td>[0.001,1000]</td><td>true</td><td>134.5075</td></tr><tr><td>Boosting type Learning rate</td><td>const.</td><td>-</td><td>gbdt</td></tr><tr><td></td><td>[0.001, 0.1]</td><td>true</td><td>0.0218</td></tr><tr><td>Num.estimators</td><td>[16,128]</td><td>true</td><td>116</td></tr><tr><td>Min. samples split. Min. samples leaf</td><td>[2,20]</td><td>false</td><td>2</td></tr><tr><td>Forest</td><td>[1,20]</td><td>false</td><td>2</td></tr><tr><td rowspan="8">e-SVR</td><td>Max.features</td><td>[0.1, 1.0]</td><td>false</td><td>0.1706</td></tr><tr><td>Bootstrap</td><td>[true, false]</td><td></td><td>false</td></tr><tr><td>C</td><td>[1.0,20.0]</td><td>true</td><td>3.066</td></tr><tr><td>coef.0 degree</td><td>[-0.5,0.5] [1,128]</td><td>false</td><td>0.1627 1</td></tr><tr><td>epsilon</td><td>[0.01,0.99]</td><td>true true</td><td>0.0251</td></tr><tr><td>gamma</td><td>[scale,auto]</td><td>=</td><td>auto</td></tr><tr><td>kernel</td><td>[linear,rbf,poly,sigmoid]</td><td></td><td>sigmoid</td></tr><tr><td>shrinking</td><td>[true, false]</td><td></td><td>false</td></tr><tr><td rowspan="10">μ-SVR</td><td>tol</td><td></td><td></td><td>0.0021</td></tr><tr><td></td><td>[0.0001,0.01]</td><td>-</td><td></td></tr><tr><td>C</td><td>[1.0,20.0]</td><td>true</td><td>5.3131</td></tr><tr><td>coef. 0</td><td>[-0.5,0.5]</td><td>false</td><td>-0.3316</td></tr><tr><td>degree</td><td>[1,128]</td><td>true</td><td>128</td></tr><tr><td>gamma</td><td>[scale,auto]</td><td></td><td>scale</td></tr><tr><td>kernel</td><td>[linear,rbf,poly,sigmoid]</td><td></td><td>rbf</td></tr><tr><td>nu</td><td>[0.01, 1.0]</td><td>false</td><td>0.1839</td></tr><tr><td>shrinking</td><td>[true, false]</td><td></td><td></td></tr><tr><td>tol</td><td>[0.0001,0.01]</td><td>=</td><td>true 0.003</td></tr></table>
+
+XGB captures the slight performance improvement of using a few skip connections better. LGB failed to capture this trend but performed very similarly to XGB for the high number of parameterfree operations.
+
+![](images/66fec0b26b6ce95ea074f449471eb035f59a7cef953258d80e01cf33be8fbbf7.jpg)  
+Figure 18: (Left) Distribution of validation error in dependence of the number of parameter-free operations in the normal cell on the NAS-Bench-301 dataset. (Middle and Right) Predictions of the GIN and XGB surrogate model. The collected groundtruth data is shown as scatter plot. Violin plots are cut off at the respective observed minimum and maximum value.
+
+# D.7 CELL TOPOLOGY ANALYSIS
+
+Furthermore, we analyze how accurate changes in the cell topology (rather than in the operations) are modeled by the surrogates. We collected groundtruth data by evaluating all Q4k=1 $\begin{array} { r } { \prod _ { k = 1 } ^ { 4 } \frac { ( k + 1 ) k } { 2 } = 1 8 0 } \end{array}$ different cell topologies (not accounting for isomorphisms) with fixed sets of operations. We assigned the same architecture to the normal and reduction cell, to focus on the effect of the cell topology. We sampled 10 operation sets uniformly at random, leading to 1800 architectures as groundtruth for this analysis.
+
+We evaluated all architectures and group the results based on the cell depth. For each of the possible cell depths, we then computed the sparse Kendall $\tau$ rank correlation between the predicted and true validation accuracy.
+
+![](images/d9f4eaca88caf625d28cb71185c091a79d0b5011094931c8fda2d949064783ce.jpg)  
+Figure 19: Comparison between GIN, XGB and LGB in the cell topology analysis.
+
+Results Results of the cell topology analysis are shown in Figure 19. We observe that LGB slightly outperforms XGB, both of which perform better on deeper cells. The GIN performs best for the shallowest cells.
+
+![](images/2e45765fdada9286389f9c176b484565c1e70e5b498a30657c32517e22ff0095.jpg)  
+Figure 20: Ground truth (GT) and surrogate trajectories on a constrained search space where the surrogates are trained with all data, leaving out the trajectories under consideration (LOTO), and leaving out all DARTS architectures (LOOO).
+
+![](images/e6a63cbd2b540f2fbe7649d61d2b3c2a2f9910c0c0c554f796e03812f3501a4a.jpg)  
+Figure 22: Comparison between the observed true trajectory of BANANAS and RS with the surrogate benchmarks only trained on well performing regions of the space
+
+# E BENCHMARK ANALYSIS
+
+# E.1 ONE-SHOT TRAJECTORIES
+
+To obtain groundtruth trajectories for DARTS, PC-DARTS and GDAS, we performed 5 runs for each optimizer with 50 search epochs and evaluated the architecture obtained by discretizing the one-shot model at each search epoch. For DARTS, in addition to the default search space, we collected trajectories on the constrained search spaces from Zela et al. (2020a) to cover a failure case where DARTS diverges and finds architectures that only contain skip connections in the normal cell. To show that our benchmark is able to predict this divergent behavior, we show surrogate trajectories when training on all data, when leaving out the trajectories under consideration from the training data, and when leaving out all DARTS data in Figure 20.
+
+While the surrogates model the divergence in all cases, they still overpredict the architectures with only skip connections in the normal cell especially when leaving out all data from DARTS. The bad performance of these architectures is predicted more accurately when including data from other DARTS runs. This can be attributed to the fact that the surrogate models have not seen any, respectively very few data, in this region of the search space. Nevertheless, it is modeled as a badperforming region and we expect that this could be further improved on by including additional training data accordingly, since including all data in training shows that the models are capable to of capturing this behavior.
+
+# E.2 ABLATION STUDY: FITTING SURROGATE MODELS ONLY ON WELL-PERFORMING REGIONS OF THE SEARCH SPACE
+
+To assess whether poorly-performing architectures are important for the surrogate benchmark, we fitted a GIN ensemble and an XGB ensemble model only on architectures that achieved a validation accuracy above $92 \%$ . We then tested on all architectures that achieved a validation below $92 \%$ .
+
+Indeed, we observe that the resulting surrogate model overpredicts accuracy in regions of the space with poor performance, resulting in a low $R ^ { 2 }$ of -0.142 and sparse Kendall tau of 0.293 for the GIN. The results for one member of the GIN ensemble are shown in Figure 21. The XGB model achieved similar results. Next, to study whether these weaker surrogate models can still be used to benchmark NAS optimizers, we also studied optimization trajectories of NAS optimizers on surrogate benchmarks based on these surrogate models. Figure 22 shows that these surrogate models indeed suffice to accurately predict the performance achieved by Random Search and BANANAS as a function of time.
+
+![](images/e5631edca0e294f45b3b659735457a4db44599438ea2f6702e62c8d2197bb204.jpg)  
+Figure 21: Scatter plot of GIN predictions on architectures that achieved below $92 \%$ validation accuracy.
+
+E.3 ABLATION STUDY: FITTING SURROGATE MODELS ONLY WITH RANDOM DATA
+
+In this section, we would like to take the Leave-One-Optimizer-Out analysis from Section 5.1 one step further by leaving out all architectures that were collected from NAS optimizers other than random search. While the LOOO analysis removes some “bias” from the benchmark (“bias” referring to its precision in a subspace), there still is the possibility that different optimizers we used explore similar subspaces, and leaving out one of them still yields “bias” induced by architectures from a similar optimizer used for generating training data. For instance, the t-SNE analysis from Figure 9 suggests that some optimizers exploit very distinct regions (e.g., BANANAS and DE) while others exploit regions somewhat similar to others (e.g., RE and PC-DARTS). The exploration behavior, on the other hand, is quite similar across optimizers since most of them perform random sampling in the beginning. Thus, in the following, we investigate whether we can create a benchmark that has no prior information about solutions any optimizer might find.
+
+To that end, we studied surrogate models based i) only on the 23746 architectures explored by random search and ii) only on 23 746 $( 4 7 . 3 \% )$ architectures of the original training set (sampled in a stratified manner, i.e., using $4 7 . 3 \%$ of the architectures from each of our sources of architectures).
+
+First, we investigated the difference in the predictive performance of surrogates based on these two different types of architectures. Specifically, we fitted our GNN and XGB surrogate models on different subsets of the respective training sets and assess their predictions on unseen architectures from all optimziers as a test set. Figure 23 shows that including architectures from optimizer trajectories in the training set consistently yields significantly better generalization.
+
+Next, we also studied the usefulness of surrogate benchmarks based on the 23 746 random architectures, compared to surrogate benchmarks based on the 23 746 architectures sampled in a stratified manner from the original set of architectures. Specifically, we used them to assess the best performance achieved by various NAS optimizers as a function of time. Comparing the trajectories in Figure 24 (based on purely random architectures for training) and Figure 25 (based on 23 746 architectures sampled in a stratified manner), we find that the surrogates fitted only on random architectures work just as well for this task as the surrogates that use architectures from NAS optimizers in their training set.
+
+Given this positive result for surrogates based purely on random architectures, we conclude that it is indeed possible to create surrogate NAS benchmarks that are by design free of bias towards any particular NAS optimizer (other than random search). While the inclusion of architectures generated with NAS optimizers in the training set substantially improves performance predictions of individual architectures, realistic trajectories of incumbent performance as a function of time can also be obtained with surrogate benchmarks based solely on random architectures. We note that the “unbiased” benchmark could possibly be further improved by utilizing more sophisticated spacefilling sampling methods, such as the ones mentioned in Appendix C.2, or by deploying surrogate models that extrapolate well.
+
+# F GUIDELINES FOR CREATING SURROGATE BENCHMARKS
+
+In order to help with the design of realistic surrogate benchmarks in the future, we provide the following list of guidelines:
+
+• Data Collection: The data collected for the NAS benchmark should provide (1) a good overall coverage, (2) explore strong regions of the space well, and (3) optimally also cover special areas in which poor generalization performance may otherwise be expected. We would like to stress that depending on the search space, a good overall coverage may already be sufficient to correctly assess the ranking of different optimizers, but as shown in Appendix E.3 additional architectures from strong regions of the space allow to increase the fidelity of the surrogate model. 1. A good overall coverage can be obtained by random search (as in our case), but one could also imagine using better space-filling designs or adaptive methods for covering the space even better. In order to add additional varied architectures, one could also think about fitting one or more surrogate models to the data collected thus far, finding the regions of maximal predicted uncertainty, evaluate architectures there and add them to the collected data, and iterate. This would constitute an active learning approach.
+
+![](images/df319399ac5fd2edf242230c6c95ffb64c77931febaafc101eb53be1d9e2ccd9.jpg)  
+Figure 23: Scatter plots of the predicted performance against the true performance of the GNN GIN/XGB surrogate models trained with different ratios of training data. ”RS” indicates that the training set only includes architectures from random search, ”mixed” indicates the training set includes architectures from all optimizers. Training set sizes are identical for the two cases. The test set contains architectures from all optimizers. For better display, we show 1000 randomly sampled architectures (blue) and 1000 architectures sampled from the top 1000 architectures (orange). For each case we also show the $R ^ { 2 }$ and Kendall- $\tau$ coefficients on the whole test set.
+
+![](images/55732f349a182356d1858c5eb093a995f9a4f400a5e20181e3913f3cf7511b52.jpg)  
+Figure 24: Anytime performance of different optimizers on the real benchmark (left) and the surrogate benchmark (GIN (middle) and XGB (right)) when training ensembles only on data collected by random search. Trajectories on the surrogate benchmark are averaged over 5 optimizer runs.
+
+![](images/f085b75340ef0af75e876fc05cf02abf36607ce8445352e847a0913ab98d99d8.jpg)  
+Figure 25: Anytime performance of different optimizers on the real benchmark (left) and the surrogate benchmark (GIN (middle) and XGB (right)) when training ensembles on $4 7 . 3 \%$ of the data collected from all optimizers. Trajectories on the surrogate benchmark are averaged over 5 optimizer runs.
+
+2. A convenient and efficient way to identify regions of strong architectures is to run NAS methods. In this case, the found regions should not only be based on the strong architectures one NAS method finds but rather on a set of strong and varied NAS methods (such as, in our case, one-shot methods and different types of discrete methods, such as Bayesian optimization and evolution). In order to add additional strong architectures, one could also think about fitting one or more several surrogate models to the data collected thus far, finding the predicted optima of these models, evaluate and add them to the collected data and iterate. This would constitute a special type of Bayesian optimization.
+
+3. Special areas in which poor generalization performance may otherwise be expected may, as in our case, e.g., include architectures with many parameterless connections, and in particular, skip connections. Other types of failure modes the community learns about would also be useful to cover.
+
+• Surrogate Models: As mentioned in the guidelines for using a surrogate benchmark (see Section 7), benchmarking an algorithm that internally uses the same model type as the surrogate model should be avoided. Therefore, to provide a benchmark for a diverse set of algorithms, we recommend providing different types of surrogate models with a surrogate benchmark. Also, in order to guard against a possible case of “bias” in a surrogate benchmark (in the sense of making more accurate predictions for architectures explored by a particular type of NAS optimizer), we recommend to provide two versions of a surrogate: one based on all available training architectures (including those found by NAS optimizers), and one based only on the data gathered for overall coverage (1. above).
+
+• Verification: As a means to verify surrogate models, we stress the importance of leave-oneoptimizer-out experiments both for data fit and benchmarking, which simulate the benchmarking of ’unseen’ optimizers.
+
+• Since most surrogate benchmarks will continue to grow for some time after their first release, to allow apples-to-apples comparisons, we strongly encourage to only release surrogate benchmarks with a version number.   
+• In order to allow the evaluation of multi-objective NAS methods, we encourage the logging of as many relevant metrics of the evaluated architectures other than accuracy as possible, including training time, number of parameters, and multiply-adds.   
+• Alongside a released surrogate benchmark, we strongly encourage to release the training data its surrogate(s) were constructed on, as well as the test data used to validate it.   
+• In order to facilitate checking hypotheses gained using the surrogate benchmarks in real experiments, the complete source code for training the architectures should be open-sourced alongside the repository, allowing to easily go back and forth between querying the model and gathering new data.

parse/train/AFiH_CNnVhS/AFiH_CNnVhS.md

@@ -0,0 +1,273 @@
+# No Fear of Heterogeneity: Classifier Calibration for Federated Learning with Non-IID Data
+
+Mi Luo1, Fei Chen2, Dapeng $\mathbf { H } \mathbf { u } ^ { 1 }$ , Yifan Zhang1, Jian Liang∗3, Jiashi Feng∗1
+
+1National University of Singapore 2Huawei Noah’s Ark Lab   
+3Institute of Automation, Chinese Academy of Sciences (CAS)
+
+{romyluo7, liangjian92, jshfeng}@gmail.com chen.f@huawei.com, {dapeng.hu, yifan.zhang}@u.nus.edu
+
+# Abstract
+
+A central challenge in training classification models in the real-world federated system is learning with non-IID data. To cope with this, most of the existing works involve enforcing regularization in local optimization or improving the model aggregation scheme at the server. Other works also share public datasets or synthesized samples to supplement the training of under-represented classes or introduce a certain level of personalization. Though effective, they lack a deep understanding of how the data heterogeneity affects each layer of a deep classification model. In this paper, we bridge this gap by performing an experimental analysis of the representations learned by different layers. Our observations are surprising: (1) there exists a greater bias in the classifier than other layers, and (2) the classification performance can be significantly improved by post-calibrating the classifier after federated training. Motivated by the above findings, we propose a novel and simple algorithm called Classifier Calibration with Virtual Representations (CCVR), which adjusts the classifier using virtual representations sampled from an approximated gaussian mixture model. Experimental results demonstrate that CCVR achieves state-of-the-art performance on popular federated learning benchmarks including CIFAR-10, CIFAR-100, and CINIC-10. We hope that our simple yet effective method can shed some light on the future research of federated learning with non-IID data.
+
+# 1 Introduction
+
+The rapid advances in deep learning have benefited a lot from large datasets like [1]. However, in the real world, data may be distributed on numerous mobile devices and the Internet of Things (IoT), requiring decentralized training of deep networks. Driven by such realistic needs, federated learning [2, 3, 4] has become an emerging research topic where the model training is pushed to a large number of edge clients and the raw data never leave local devices.
+
+A notorious trap in federated learning is training with non-IID data. Due to diverse user behaviors, large heterogeneity may be present in different clients’ local data, which has been found to result in unstable and slow convergence [5] and cause suboptimal or even detrimental model performance [6, 7]. There have been a plethora of works exploring promising solutions to federated learning on non-IID data. They can be roughly divided into four categories: 1) client drift mitigation [5, 8, 9, 10], which modifies the local objectives of the clients, so that the local model is consistent with the global model to a certain degree; 2) aggregation scheme [11, 12, 13, 14, 15], which improves the model fusion mechanism at the server; 3) data sharing [6, 16, 17, 18], which introduces public datasets or synthesized data to help construct a more balanced data distribution on the client or on the server;
+
+4) personalized federated learning [19, 20, 21, 22], which aims to train personalized models for individual clients rather than a shared global model.
+
+However, as suggested by [7], existing algorithms are still unable to achieve good performance on image datasets with deep learning models, and could be no better than vanilla FedAvg [2]. To identify the reasons behind this, we perform a thorough experimental investigation on each layer of a deep neural network. Specifically, we measure the Centered Kernel Alignment (CKA) [23] similarity between the representations from the same layer of different clients’ local models. The observation is thought-provoking: comparing different layers learned on different clients, the classifier has the lowest feature2 similarity across different local models.
+
+Motivated by the above discovery, we dig deeper to study the variation of the weight of the classifier in federated optimization, and confirm that the classifier tends to be biased to certain classes. After identifying this devil, we conduct several empirical trials to debias the classifier via regularizing the classifier during training or calibrating classifier weights after training. We surprisingly find that post-calibration strategy is particularly useful — with only a small fraction of IID data, the classification accuracy is significantly improved. However, this approach cannot be directly deployed in practice since it infringes the privacy rule in federated learning.
+
+Based on the above findings and considerations, we propose a novel and privacy-preserving approach called Classifier Calibration with Virtual Representations (CCVR) which rectifies the decision boundaries (the classifier) of the deep network after federated training. CCVR generates virtual representations based on an approximated Gaussian Mixture Model (GMM) in the feature space with the learned feature extractor. Experimental results show that CCVR achieves significant accuracy improvements over several popular federated learning algorithms, setting the new state-of-the-art on common federated learning benchmarks like CIFAR-10, CIFAR-100 and CINIC-10.
+
+To summarize, our contributions are threefold: (1) We present the first systematic study on the hidden representations of different layers of neural networks (NN) trained with FedAvg on non-IID data and provide a new perspective of understanding federated learning with heterogeneous data. (2) Our study reveals an intriguing fact that the primary reason for the performance degradation of NN trained on non-IID data is the classifier. (3) We propose CCVR (Classifier Calibration with Virtual Representations) — a simple and universal classifier calibration algorithm for federated learning. CCVR is built on top of the off-the-shelf feature extractor and requires no transmission of the representations of the original data, thus raising no additional privacy concern. Our empirical results show that CCVR brings considerable accuracy gains over vanilla federated learning approaches.
+
+# 2 Related Work
+
+Federated learning [2, 3, 4] is a fast-growing research field and remains many open problems to solve. In this work, we focus on addressing the non-IID quagmire [6, 24]. Relevant works have pursued the following four directions.
+
+Client Drift Mitigation. FedAvg [2] has been the de facto optimization method in the federated setting. However, when it is applied to the heterogeneous setting, one key issue arises: when the global model is optimized with different local objectives with local optimums far away from each other, the average of the resultant client updates (the server update) would move away from the true global optimum [9]. The cause of this inconsistency is called ‘client drift’. To alleviate it, FedAvg is compelled to use a small learning rate which may damage convergence, or reduce the number of local iterations which induces significant communication cost [25]. There have been a number of works trying to mitigate ‘client drift’ of FedAvg from various perspectives. FedProx [5] proposes to add a proximal term to the local objective which regularizes the euclidean distance between the local model and the global model. MOON [8] adopts the contrastive loss to maximize the agreement of the representation learned by the local model and that by the global model. SCAFFOLD [9] performs ‘client-variance reduction’ and corrects the drift in the local updates by introducing control variates. FedDyn [10] dynamically changes the local objectives at each communication round to ensure that the local optimum is asymptotically consistent with the stationary points of the global objective. FedIR [26] applies importance weight to the local objective, which alleviates the imbalance caused by non-identical class distributions among clients.
+
+![](images/f73990a9b9f0a9deb71881aa7541b9f5d00d4c6813cb174aa1f1509b33573888.jpg)  
+Figure 1: CKA similarities of three different layers of different ‘client model-client model’ pairs.
+
+![](images/d4cfbd236c9649235806ba08291299e2f86a36d71753e6483e99a39a2f715ec4.jpg)  
+Figure 2: The means of the CKA similarities of different layers in different local models.
+
+Aggregation Scheme. A fruitful avenue of explorations involves improvements at the model aggregation stage. These works are motivated by three emerging concerns. First, oscillation may occur when updating the global model using gradients collected from clients with a limited subset of labels. To alleviate it, [11] proposes FedAvgM which adopts momentum update on the server-side. Second, element-wise averaging of weights may have drastic negative effects on the performance of the averaged model. [12] shows that directly averaging local models that are learned from totally distinct data distributions cannot produce a global model that performs well on the global distribution. The authors further propose FedDF that leverages unlabeled data or artificial samples generated by GANs [27] to distill knowledge from the local models. [13] considers the setting where each client performs variable amounts of local works and proposes FedNova which normalizes the local updates before averaging. Third, a handful of works [14, 15] believe that the permutation invariance of neural network parameters may cause neuron mismatching when conducting coordinate-wise averaging of model weights. So they propose to match the parameters of local models while aggregating.
+
+Data Sharing. The key motivation behind data sharing is that a client cannot acquire samples from other clients during local training, thus the learned local model under-represents certain patterns or samples from the absent classes. The common practices are to share a public dataset [6], synthesized data [16, 17] or a condensed version of the training samples [18] to supplement training on the clients or on the server. This line of works may violate the privacy rule of federated learning since they all consider sharing raw input data of the model, either real data or artificial data.
+
+Personalized Federated Learning. Different from the above directions that aim to learn a single global model, another line of research focuses on learning personalized models. Several works aim to make the global model customized to suit the need of individual users, either by treating each client as a task in meta-learning [19, 28, 20, 29] or multi-task learning [30], or by learning both global parameters for all clients and local private parameters for individual clients [21, 31, 32]. There are also heuristic approaches that divide clients into different clusters based on their learning tasks (objectives) and perform aggregation only within the cluster [33, 34, 22, 35].
+
+In this work, we consider training a single global classification model. To the best of our knowledge, we are the first to decouple the representation and classifier in federated learning — calibrating classifier after feature learning. Strictly speaking, our proposed CCVR algorithm does not fall into any aforementioned research direction but can be readily combined with most of the existing federated learning approaches to achieve better classification performance.
+
+# 3 Heterogeneity in Federated Learning: The Devil Is in Classifier
+
+# 3.1 Problem Setup
+
+We aim to collaboratively train an image classification model in a federated learning system which consists of $K$ clients indexed by $[ K ]$ and a central server. Client $k$ has a local dataset $\mathcal { D } ^ { \bar { k } }$ , and we set $\textstyle { \mathcal { D } } = \bigcup _ { k \in [ K ] } { \mathcal { D } } ^ { k }$ as the whole dataset. Suppose there are $C$ classes in $\mathcal { D }$ indexed by $[ C ]$ . Denote by $( \pmb { x } , y ) \in \mathcal { X } \times [ C ]$ a sample in $\mathcal { D }$ , where $_ { \textbf { \em x } }$ is an image in the input space $\mathcal { X }$ and $y$ is its corresponding label. Let $\mathcal { D } _ { c } ^ { k } = \{ ( x , y ) \in \mathcal { D } ^ { k } : y = c \}$ be the set of samples with ground-truth label $c$ on client $k$ . We decompose the classification model into a deep feature extractor and a linear classifier. Given a sample $( { \pmb x } , y )$ , the feature extractor $f _ { \pmb \theta } : \mathcal { X }  \mathcal { Z }$ , parameterized by $\pmb \theta$ , maps the input image $_ { \textbf { \em x } }$ into a feature vector $z = f _ { \pmb \theta } ( \pmb x ) \in \mathbb R ^ { d }$ in the feature space $\mathcal { Z }$ . Then the classifier $g _ { \varphi } : \mathcal { Z } \to \mathbb { R } ^ { C }$ , parameterized by $\varphi$ , produces a probability distribution $g _ { \varphi } ( z )$ as the prediction for $_ { \textbf { \em x } }$ . Denote by ${ \pmb w } = ( { \pmb \theta } , \varphi )$ the parameter of the classification model.
+
+![](images/7d66b323c3a2a81b5e25e260d614d1f83c3e698101f31bbda35e9d4dfd4813ed.jpg)  
+Figure 3: Label distribution of CIFAR-10 across clients (the first graph) and the classifier weight norm distribution across clients in different rounds and data partitions (the three graphs on the right).
+
+Federated learning proceeds through the communication between clients and the server in a roundby-round manner. In round $t$ of the process, the server sends the current model parameter $\pmb { w } ^ { ( t - 1 ) }$ to a set $U ^ { ( t ) }$ of selected clients. Then each client $k \in U ^ { ( t ) }$ locally updates the received parameter $\pmb { w } ^ { ( t - 1 ) }$ to wk ${ \pmb w } _ { k } ^ { ( t ) }$ with the following objective:
+
+$$
+\operatorname* { m i n } _ { \pmb { w } _ { k } ^ { ( t ) } } \mathbb { E } _ { ( \pmb { x } , y ) \sim \mathcal { D } ^ { k } } [ \mathcal { L } ( \pmb { w } _ { k } ^ { ( t ) } ; \pmb { w } ^ { ( t - 1 ) } , \pmb { x } , y ) ] ,
+$$
+
+where $\mathcal { L }$ is the loss function. Note that $\mathcal { L }$ is algorithm-dependent and could rely on the current global model parameter for a number of e $w ^ { ( t - 1 ) }$ as well. For instance, FedAvg [2] computes sing the cross-entropy loss, with initialization $\boldsymbol { w } _ { k } ^ { ( t ) }$ by running SG parameter set to $\mathcal { D } ^ { k }$ $w ^ { ( t - 1 ) }$ FedProx [5] uses the cross entropy loss with an $L _ { 2 }$ -regularization term to constrain the distance between $w _ { k } ^ { ( t ) }$ and $\pmb { w } ^ { ( t - 1 ) }$ ; MOON [8] introduces a contrastive loss term to address the feature drift issue. In the end of round $t$ , the selected clients send the optimized parameter back to the server and the server updates the parameter by aggregating heterogeneous parameters as follows,
+
+$$
+\pmb { w } ^ { ( t ) } = \sum _ { k \in U ^ { ( t ) } } p _ { k } \pmb { w } _ { k } ^ { ( t ) } , \mathrm { ~ w h e r e ~ } p _ { k } = \frac { | \mathcal { D } ^ { k } | } { \sum _ { k ^ { \prime } \in U ^ { ( t ) } } | \mathcal { D } ^ { k ^ { \prime } } | } .
+$$
+
+# 3.2 A Closer Look at Classification Model: Classifier Bias
+
+To vividly understand how non-IID data affect the classification model in federated learning, we perform an experimental study on heterogeneous local models. For the sake of simplicity, we choose CIFAR-10 with 10 clients which is a standard federated learning benchmark, and a convolutional neural network with 7 layers used in [8]. As for the non-IID experiments, we partition the data according to the Dirichlet distribution with the concentration parameter $\alpha$ set as 0.1. More details are covered in the Appendix. To be specific, for each layer in the model, we leverage the recently proposed Centered Kernel Alignment (CKA) [23] to measure the similarity of the output features between two local models, given the same input testing samples. CKA outputs a similarity score between 0 (not similar at all) and 1 (identical). We train the model with FedAvg for 100 communication rounds and each client optimizes for 10 local epochs at each round.
+
+We first selectively show the pairwise CKA features similarity of three different layers across local models in Figure 1. Three compared layers here are the first layer, the middle layer (Layer 4), and the last layer (the classifier), respectively. Interestingly, we find that features outputted by the deeper layer show lower CKA similarity. It indicates that, for federated models trained on non-IID data, the deeper layers have heavier heterogeneity across different clients. By averaging the pairwise CKA features similarity in Figure 1, we can obtain a single value to approximately represent the similarity of the feature outputs by each layer across different clients. We illustrate the approximated layer-wise features similarity in Figure 2. The results show that the models trained with non-IID data have consistently lower feature similarity across clients for all layers, compared with those trained on IID data. The primary finding is that, for non-IID training, the classifier shows the lowest features similarities, among all the layers. The low CKA similarities of the classifiers imply that the local classifiers change greatly to fit the local data distribution.
+
+Table 1: Accuracy $@ 1$ $( \% )$ on CIFAR-10 with different degrees of heterogeneity.   
+
+<table><tr><td>Method</td><td>α= 0.5</td><td>α= 0.1</td><td>α=0.05</td></tr><tr><td>FedAvg</td><td>68.62±0.77</td><td>58.55±0.98</td><td>52.33±0.43</td></tr><tr><td>FedAvg + clsnorm</td><td>69.65±0.35 (↑ 1.03)</td><td>58.94±0.08 (↑ 0.39)</td><td>51.74±4.02 (↓ 0.59)</td></tr><tr><td>FedAvg +clsprox</td><td>68.82±0.75 (↑ 0.20)</td><td>59.04±0.70 (↑ 0.49)</td><td>52.38±0.78( (↑0.05)</td></tr><tr><td>FedAvg + clsnorm + clsprox</td><td>68.75±0.75 (↑ 0.13)</td><td>58.80±0.30 (↑ 0.25)</td><td>52.39±0.24 (↑ 0.06)</td></tr><tr><td>FedAvg + calibration (whole data)</td><td>72.51±0.53 (↑ 3.89)</td><td>64.70±0.94 (↑ 6.15)</td><td>57.53±1.00 (↑ 5.20)</td></tr></table>
+
+![](images/f06e26eac31c1eb1a9feb2f784cdfb9fee2a8da8ca825a06fb861cbae732b8b0.jpg)  
+Figure 4: The effect of classifier calibration using different amounts of data.
+
+To perform a deeper analysis on the classifier trained on non-IID data, inspired by [36], we illustrate the $L _ { 2 }$ norm of the local classifier weight vectors in Figure 3. We observe that the classifier weight norms would be biased to the class with more training samples at the initial training stage. At the end of the training, models trained on non-IID data suffer from a much heavier biased classifier than the models trained on IID data.
+
+Based on the above observations about the classifier, we hypothesize that: because the classifier is the closest layer to the local label distribution, it can be easily biased to the heterogeneous local data, reflected by the low features similarity among different local classifiers and the biased weight norms. Furthermore, we believe that debiasing the classifier is promising to directly improve the classification performance.
+
+# 3.3 Classifier Regularization and Calibration
+
+To effectively debias the classifier, we consider the following regularization and calibration methods.
+
+Classifier Weight $L 2$ -normalization. To eliminate the bias in classifier weight norms, we normalize the classifier weight vectors during the training and the inference stage. We abbreviate it to ‘clsnorm’. In particular, the classifier is a linear transformation with weight $\bar { \boldsymbol { \varphi } } = [ \varphi _ { 1 } , \ldots , \varphi _ { C } ]$ , followed by normalization and softmax. Given a feature $_ { z }$ , the output of the classifier is
+
+$$
+g _ { \varphi } ( z ) _ { i } = \frac { e ^ { \varphi _ { i } ^ { T } z / | | \varphi _ { i } | | } } { \sum _ { i ^ { \prime } = 1 } ^ { C } e ^ { \varphi _ { i ^ { \prime } } ^ { T } z / | | \varphi _ { i ^ { \prime } } | | } } , \quad \forall i \in [ C ] .
+$$
+
+Classifier Quadratic Regularization. Beyond restricting the weight norms of classifier, we also consider adding a proximal term similar to [5] only to restrict the classifier weights to be close to the received global classifier weight vectors from the server. We write it as ‘clsprox’ for short. The loss function in Eq. (1) can be specified as
+
+$$
+\mathcal { L } ( \boldsymbol { w } _ { k } ^ { ( t ) } ; \boldsymbol { w } ^ { ( t - 1 ) } , \boldsymbol { x } , \boldsymbol { y } ) = \ell ( g _ { \varphi _ { k } ^ { ( t ) } } ( f _ { \theta _ { k } ^ { ( t ) } } ( \boldsymbol { x } ) ) , \boldsymbol { y } ) + \frac { \mu } { 2 } | | \varphi _ { k } ^ { ( t ) } - \varphi ^ { ( t - 1 ) } | | ^ { 2 } ,
+$$
+
+where $\ell$ is the cross-entropy loss and $\mu$ is the regularization factor.
+
+Classifier Post-calibration with IID Samples. In addition to regularizing the classifier during federated training, we also consider a post-processing technique to adjust the learned classifier. After the federated training, we fix the feature extractor and calibrate the classifier by SGD optimization with a cross-entropy loss on IID samples. Note that this calibration strategy requires IID raw features collected from heterogeneous clients. Therefore, it can only serve as an experimental study use but cannot be applied to the real federated learning system.
+
+We conduct experiments to compare the above three methods on CIFAR-10 with three different degrees of data heterogeneity and present the results in Table 1. We observe that regularizing the L2-norm of classifier weight (clsnorm) is effective for light data heterogeneity but would have less help or even lead to damages along with the increase of the heterogeneity. Regularizing the classifier parameters (clsprox) is consistently effective but with especially minor improvements. Surprisingly, we find that calibrating the classifier of the FedAvg model with all training samples brings significant performance improvement for all degrees of data heterogeneity.
+
+To further understand the classifier calibration technique, we additionally perform calibrations with different numbers of data samples and different off-the-shelf federated models trained by FedAvg and FedProx. The results are shown in Figure 4 and we observe that data-based classifier calibration performs consistently well, even with $1 / 5 0$ training data samples for calibration use. These significant performance improvements after adjusting the classifier strongly verify our aforementioned hypothesis, i.e., the devil is in the classifier.
+
+# 4 Classifier Calibration with Virtual Representations
+
+Motivated by the above observations, we propose Classifier Calibration with Virtual Representations (CCVR) that runs on the server after federated training the global model. CCVR uses virtual features drawn from an estimated Gaussian Mixture Model (GMM), without accessing any real images. Suppose $f _ { \widehat { \pmb { \theta } } }$ and $g _ { \widehat { \varphi } }$ b bare the feature extractor and classifier of the global model, respectively, where $\widehat { \pmb { w } } = ( \widehat { \pmb { \theta } } , \widehat { \pmb { \varphi } } )$ b bis the parameter trained by a certain federated learning algorithm, e.g. FedAvg. We shall use $f _ { \widehat { \pmb { \theta } } }$ to extract features and estimate the correbsponding feature distribution, and re-train $g$ using generated virtual representations.
+
+Feature Distribution Estimation. For semantics related tasks such as classification, the features learned by deep neural networks can be approximated with a mixture of Gaussian distribution. Theoretically, any continuous distribution can be approximated by using a finite number of mixture of gaussian distributions [37]. In our CCVR, we assume that features of each class in $\mathcal { D }$ follow a Gaussian distribution. The server estimates this distribu
+
+# Algorithm 1: Virtual Representation Generation
+
+Input: Feature extractor $f _ { \widehat { \pmb { \theta } } }$ of the global model, number $M _ { c }$ bof virtual features for class $c$ 1 # Server executes:   
+2 Send $f _ { \widehat { \pmb { \theta } } }$ to clients. 3 # Clients execute: 4 foreach client $k \in [ K ]$ do 5 foreach class $c \in [ C ]$ do 6 Produce ${ \ z } _ { c , k , j } = \bar { \ z } _ { \widehat { \theta } } ( \pmb { x } _ { c , k , j } )$ for $j$ -th sample in $\mathcal { D } _ { c } ^ { k }$ for $j \in [ N _ { c , k } ]$ . 7 Compute $\mu _ { c , k }$ and $\Sigma _ { c , k }$ using Eq. (2). 8 end 9 Send $\{ ( \pmb { \mu _ { c , k } } , \pmb { \Sigma _ { c , k } } ) : c \in [ C ] \}$ to server.   
+10 end   
+11 # Server executes:   
+12 foreach class $c \in [ C ]$ do   
+13 Compute $\pmb { \mu } _ { c }$ and $\Sigma _ { c }$ using Eq. (3) and (4).   
+14 Draw a set $G _ { c }$ of $M _ { c }$ features from $\mathcal { N } ( \mu _ { c } , \Sigma _ { c } )$ with ground truth label $c$ .   
+15 end   
+Output: Set of virtual representations $\cup _ { c \in [ C ] } G _ { c }$
+
+tion by computing the mean $\pmb { \mu } _ { c }$ and the covariance $\Sigma _ { c }$ for each class $c$ of $\mathcal { D }$ using gathered local statistics from clients, without accessing true data samples or their features. In particular, the server first sends the feature extractor $f _ { \widehat { \pmb { \theta } } }$ of the trained global model to clients. Let $\dot { N } _ { c , k } = | \mathcal { D } _ { c } ^ { k } |$ be the number of samples of class $c$ θb on client $k$ , and set $\begin{array} { r } { N _ { c } = \sum _ { k = 1 } ^ { K } N _ { c , k } } \end{array}$ . Client $k$ produces features $\{ z _ { c , k , 1 } , \dots , z _ { c , k , N _ { c , k } } \}$ for class $c$ , where $\boldsymbol { z } _ { c , k , j } = f _ { \widehat { \theta } } ( \boldsymbol { x } _ { c , k , j } )$ is the feature of the $j$ -th sample in $\mathcal { D } _ { c } ^ { k }$ , and computes local mean $\mu _ { c , k }$ and covariance $\Sigma _ { c , k }$ of $\mathcal { D } _ { c } ^ { k }$ as:
+
+$$
+\boldsymbol { \mu } _ { c , k } = \frac { 1 } { N _ { c , k } } \sum _ { j = 1 } ^ { N _ { c , k } } z _ { c , k , j } , \quad \boldsymbol { \Sigma } _ { c , k } = \frac { 1 } { N _ { c , k } - 1 } \sum _ { j = 1 } ^ { N _ { c , k } } \left( z _ { c , k , j } - \mu _ { c , k } \right) \left( z _ { c , k , j } - \mu _ { c , k } \right) ^ { T } ,
+$$
+
+Then client $k$ uploads $\{ ( \pmb { \mu _ { c , k } } , \pmb { \Sigma _ { c , k } } ) : c \in [ C ] \}$ to server. For the server to compute the global statistics of $\mathcal { D }$ , it is sufficient to represent the global mean $\pmb { \mu } _ { c }$ and covariance $\Sigma _ { c }$ using $\mu _ { c , k }$ ’s and $\Sigma _ { c , k }$ ’s for each class $c$ . The global mean can be straightforwardly written as
+
+$$
+\pmb { \mu } _ { c } = \frac { 1 } { N _ { c } } \sum _ { k = 1 } ^ { K } \sum _ { j = 1 } ^ { N _ { c , k } } z _ { c , k , j } = \sum _ { k = 1 } ^ { K } \frac { N _ { c , k } } { N _ { c } } \pmb { \mu } _ { c , k } .
+$$
+
+For the covariance, note that by definition we have
+
+$$
+( N _ { c , k } - 1 ) \boldsymbol { \Sigma } _ { c , k } = \sum _ { j = 1 } ^ { N _ { c , k } } z _ { c , k , j } z _ { c , k , j } ^ { T } - N _ { c , k } \cdot \mu _ { c , k } \mu _ { c , k } ^ { T }
+$$
+
+whenever $N _ { c , k } \ge 1$ . Then the global covariance can be written as
+
+$$
+\begin{array} { l } { \displaystyle \pmb { \Sigma } _ { c } = \frac { 1 } { N _ { c } - 1 } \sum _ { k = 1 } ^ { K } \sum _ { j = 1 } ^ { N _ { c , k } } z _ { c , k , j } z _ { c , k , j } ^ { T } - \frac { N _ { c } } { N _ { c } - 1 } \pmb { \mu } _ { c } \pmb { \mu } _ { c } ^ { T } } \\ { \displaystyle = \sum _ { k = 1 } ^ { K } \frac { N _ { c , k } - 1 } { N _ { c } - 1 } \pmb { \Sigma } _ { c , k } + \sum _ { k = 1 } ^ { K } \frac { N _ { c , k } } { N _ { c } - 1 } \pmb { \mu } _ { c , k } \pmb { \mu } _ { c , k } ^ { T } - \frac { N _ { c } } { N _ { c } - 1 } \pmb { \mu } _ { c } \pmb { \mu } _ { c } ^ { T } . } \end{array}
+$$
+
+Virtual Representations Generation. After obtaining $\pmb { \mu } _ { c }$ ’s and $\Sigma _ { c }$ ’s, the server generates a set $G _ { c }$ of virtual features with ground truth label $c$ from the Gaussian distribution $\mathcal { N } ( \mu _ { c } , \Sigma _ { c } )$ . The number $M _ { c } : = | G _ { c } |$ of virtual features for each class $c$ could be determined by the fraction $\frac { N _ { c } } { | \mathcal { D } | }$ to reflect the inter-class distribution. See Algorithm 1.
+
+Classifier Re-Training. The last step of our CCVR method is classifier re-training using virtual representations. We take out the classifier $g$ from the global model, initialize its parameter as $\widehat { \varphi }$ , and re-train the parameter to $\widetilde { \varphi }$ for the objective
+
+$$
+\operatorname* { m i n } _ { \tilde { \varphi } } \mathbb { E } _ { ( z , y ) \sim \bigcup _ { c \in [ C ] } G _ { c } } [ \ell ( g _ { \tilde { \varphi } } ( z ) , y ) ] ,
+$$
+
+where $\ell$ is the cross-entropy loss. We then obtain the final classification model $g _ { \widetilde { \varphi } } \circ f _ { \widehat { \theta } }$ consisting of the pre-trained feature extractor and the calibrated classifier.
+
+Privacy Protection. CCVR protects privacy at the basic level because each client only uploads their local Gaussian statistics rather than the raw representations. Note that CCVR is just a post-hoc method, so it can be easily combined with some privacy protection techniques [38] to further secure privacy. In the Appendix, we provide an empirical analysis on the privacy-preserving aspect.
+
+# 5 Experiment
+
+# 5.1 Experiment Setup
+
+Federated Simulation. We consider image classification task and adopt three datasets from the popular FedML benchmark [39], i.e., CIFAR-10 [40], CIFAR-100 [40] and CINIC-10 [41]. Note that CINIC-10 is constructed from ImageNet [42] and CIFAR-10, whose samples are very similar but not drawn from identical distributions. Therefore, it naturally introduces distribution shifts which is suited to the heterogeneous nature of federated learning. To simulate federated learning scenario, we randomly split the training set of each dataset into $K$ batches, and assign one training batch to each client. Namely, each client owns its local training set. We hold out the testing set at the server for evaluation of the classification performance of the global model. For hyperparameter tuning, we first take out a $15 \%$ subset of training set for validation. After selecting the best hyperparameter, we return the validation set to the training set and retrain the model. We are interested in the NIID partitions of the three datasets, where class proportions and number of data points of each client are unbalanced. Following [14, 15], we sample $p _ { i } \sim D i r _ { K } ( \alpha )$ and assign a $p _ { i , k }$ proportion of the samples from class $i$ to client $k$ . We set $\alpha$ as 0.5 unless otherwise specified. For fair comparison, we apply the same data augmentation techniques for all methods.
+
+Table 2: Accuracy $@ 1$ $( \% )$ on CIFAR-10 with different degrees of heterogeneity (α ∈ $\{ 0 . 5 , 0 . 1 , 0 . 0 5 \} _ { \ r }$ ), CIFAR-100 and CINIC-10.   
+
+<table><tr><td></td><td>Method</td><td>α=0.5</td><td>α=0.1</td><td>α=0.05</td><td>CIFAR-100</td><td>CINIC-10</td></tr><tr><td rowspan="4">No Calibration</td><td>FedAvg</td><td>68.62±0.77</td><td>58.55±0.98</td><td>52.33±0.43</td><td>66.25±0.54</td><td>60.20±2.04</td></tr><tr><td>FedProx</td><td>69.07±1.07</td><td>58.93±0.64</td><td>53.00±0.32</td><td>66.31±0.39</td><td>60.52±2.07</td></tr><tr><td>FedAvgM</td><td>69.00±1.68</td><td>59.22±1.14</td><td>51.98±0.91</td><td>66.43±0.23</td><td>60.46±0.73</td></tr><tr><td>MOON</td><td>70.48±0.36</td><td>57.36±0.85</td><td>49.91±0.38</td><td>67.02±0.31</td><td>65.67±2.10</td></tr><tr><td rowspan="4">CCVR (Ours.)</td><td>FedAvg</td><td></td><td></td><td>71.03±0.40(↑2.41) 62.68±0.54(个4.13) 54.95±0.61(↑ 2.62)</td><td>66.60±0.63(↑0.35)</td><td>69.99±0.54 (↑9.79)</td></tr><tr><td>FedProx</td><td>70.99±1.21(个 1.92) 62.60±0.43(↑ 3.67)</td><td></td><td>55.79±1.07 (↑ 2.79) 66.61±0.48 (↑0.30)</td><td></td><td>70.05±0.66 (↑ 9.53)</td></tr><tr><td>FedAvgM</td><td>71.49±0.88 (↑ 2.49)</td><td>62.64±1.07 (个 3.42)</td><td>54.57±0.58 (↑ 2.59)</td><td>66.71±0.16(↑0.28)</td><td>70.87±0.61 (↑ 10.41)</td></tr><tr><td>MOON</td><td>71.29±0.11 (↑ 0.81)</td><td>62.22±0.70(↑ 4.86)</td><td>55.60±0.63 (↑ 5.69)</td><td>67.17±0.37 (↑ 0.15)</td><td>69.42±0.65 (↑ 3.75)</td></tr><tr><td rowspan="4">Oracle</td><td>FedAvg</td><td>72.51±0.53 (↑ 3.89)</td><td>64.70±0.94(↑6.15)</td><td>57.53±1.00 (个5.20)</td><td>66.84±0.50(↑0.59)</td><td>73.47±0.30(个 13.27)</td></tr><tr><td>FedProx</td><td>72.26±1.22 (↑ 3.19)</td><td>64.63±0.93(↑ 5.70)</td><td>57.33±0.72 (↑4.33)</td><td>66.68±0.43 (↑0.37)</td><td>73.10±0.57 (↑ 12.58)</td></tr><tr><td>FedAvgM</td><td>73.30±0.19 (↑ 4.30)</td><td>64.24±1.32(↑ 5.02)</td><td>57.11±1.08 (↑ 5.13)</td><td>66.94±0.32 (↑ 0.51)</td><td>72.88±0.37 (↑ 12.42)</td></tr><tr><td>MOON</td><td>72.05±0.16 (↑ 1.57)</td><td>64.94±0.58 (个 7.58)</td><td>58.14±0.47 (个 8.23)</td><td>67.56±0.44 (↑ 0.54)</td><td>73.38±0.23 (↑ 7.71)</td></tr></table>
+
+Baselines and Implementation. We consider comparing the test accuracies of the representative federated learning algorithms FedAvg [2], FedProx [5], FedAvgM [11, 26] and the state-of-the-art method MOON [8] before and after applying our CCVR. For FedProx and MOON, we carefully tune the coefficient of local regularization term $\mu$ and report their best results. For FedAvgM, the server momentum is set to be 0.1. We use a simple 4-layer CNN network with a 2-layer MLP projection head described in [8] for CIFAR-10. For CIFAR-100 and CINIC-10, we adopt MobileNetV2 [43]. For CCVR, to make the virtual representations more Gaussian-like, we apply ReLU and Tukey’s transformation before classifier re-training. For Tukey’s transformation, the parameter is set to be 0.5. For each dataset, all methods are evaluated with the same model for fair comparison. The proposed CCVR algorithm only has one important hyperparameter, the number of feature samples $M _ { c }$ to generate. Unless otherwise stated, $M _ { c }$ is set to 100, 500 and 1000 for CIFAR-10, CIFAR100 and CINIC-10 respectively. All experiments run with PyTorch 1.7.1. More details about the implementation and datasets are summarized in the Appendix.
+
+# 5.2 Can classifier calibration improve performance of federated learning?
+
+In Table 2, we present the test accuracy on all datasets before and after applying our CCVR. We also report the results under an ideal setting where the whole data are available for classifier calibration (Oracle). These results indicate the upper bound of classifier calibration.
+
+CCVR consistently improves all baseline methods. First, it can be observed that applying classifier calibration increases accuracies for all baseline methods, even with the accuracy gain up to $1 0 . 4 1 \%$ on CINIC-10. This is particularly inspiring because CCVR requires no modification to the original federated training process. One can easily get considerable accuracy profits by simply post-processing the trained global model. Comparing the accuracy gains of different methods after applying CCVR and whole data calibration, we find that the accuracies of FedAvg and MOON get the greatest increase. On CINIC-10, the oracle results of FedAvg even outstrip those of all other baselines, implying that FedAvg focuses more on learning high-quality features but ignores learning a fair classifier. It further confirms the necessity of classifier calibration.
+
+# 5.3 In what situation does CCVR work best?
+
+We observe that though there is improvement on CIFAR-100 by applying CCVR, it seems subtle compared with that of other two datasets. This is not surprising, since the final accuracy achieved by classifier calibration is not only dependent on the degree to which the classifier is debaised, but also closely correlated with the quality of pre-trained representations. In CIFAR-100, each class only has 500 training images, so the classification task itself is very difficult and the model may learn representations with low separability. It is shown that the accuracy obtained with CCVR on CIFAR-100 is very close to the upper bound, indicating that CCVR does a good job of correcting the classifier, even if it is provided with a poor feature extractor.
+
+We also note that CCVR achieves huge improvements on CINIC-10. To further analyze the reason of this success and the characteristics of CCVR, we now show the t-SNE visualization [44] of the features learned by FedAvg on CINIC-10 dataset in Figure 5. From the first and second sub-graphs, we can observe that some classes dominate the classification results, while certain classes are rarely predicted correctly. For instance, the classifier makes wrong prediction for most of the samples belonging to the grey class. Another evidence showing there exists a great bias in the classifier is that, from the upper right corner of the ground truth sub-graph, we can see that the features colored green and those colored purple can be easily separated. However, due to biases in the classifier, nearly all purple features are wrongly classified as the green class. Observing the third sub-graph, we find that by applying CCVR, these misclassifications are alleviated. We also find that, with CCVR, mistakes are basically made when identifying easily-confused features that are close to the decision boundary rather than a majority of features that belong to certain classes. This suggests that the classifier weight has been adjusted to be more fair to each class. In summary, CCVR may be more effective when applied to the models with good representations but serious classifier biases.
+
+![](images/f8d5b82405ebca032c0564f13f7a2ea6db7a6f88119227b3823ff0e2ce66696f.jpg)  
+Figure 5: t-SNE visualization of the features learned by FedAvg on CINIC-10. The features are colored by the ground truth and the predictions of the classifier before and after applying CCVR. Best viewed in color.
+
+# 5.4 How to forecast the performance of classifier calibration?
+
+We resort to Sliced Wasserstein Distance [45], which is a popular metric to measure the distances between distributions, to quantify the separability of GMM. The experiments are conducted on CIFAR-10 with $\alpha = 0 . 1$ . We first compute the Wasserstein distances between any two mixtures, then we average all the distances to get a mean distance. The farther the distance, the better the separability of GMM. We visualize the relationship between the accuracy gains and the separability of GMM in Figure 6. It is observed that the mean Wasserstein distance of GMM is positively correlated with the accuracy upper bound of classifier calibration. It verifies our claim in Section 5.3: CCVR may be more effective when applied to the models with good (separable) representations. In practice, one can use the mean Wasserstein distance of GMM to evaluate the quality of the simulated representations, as well as to forecast the potential performance of classifier calibration.
+
+![](images/67b14e716f7af69fe0d9be5e2f52490fb67b213ad62a775ce51b2ba7cf1a4af5.jpg)  
+Figure 6: GMM’s separability.
+
+# 5.5 How many virtual features to generate?
+
+One important hyperparameter in our CCVR is the number of virtual features $M _ { c }$ for each class $c$ to generate. We study the effect of $M _ { c }$ by tuning it from $\{ 0 , 5 0 , 1 0 0 , 5 0 0 , 1 0 0 0 , 2 0 0 0 \}$ on three different partitions of CIFAR-10 $\alpha \in \{ 0 . 0 5 , 0 . 1 , 0 . 5 \} )$ ) when applying CCVR to FedAvg. The results are provided in Figure 7. In general, even sampling only a few features can significantly increase the classification accuracy. Additionally, it is observed that on the two more heterogeneous distributions (the left two sub-graphs), more samples produces higher accuracy. Although results on NIID-0.5 give a similar hint in general, an accuracy decline when using a medium number of virtual samples is observed. This suggests that $M _ { c }$ is more sensitive when faced with a more balanced dataset. This can be explained by the nature of CCVR: utilizing virtual feature distribution to mimic the original feature distribution. As a result, if the number of virtual samples is limited, the simulated distribution may deviates from the true feature distribution. The results on NIID-0.5 implies that this trap could be easier to trigger when CCVR dealing with a more balanced original distribution. To conclude, though CCVR can provide free lunch for federated classification, one should still be very careful when tuning $M _ { c }$ to achieve higher accuracy. Generally speaking, a larger value of $M _ { c }$ is better.
+
+![](images/9c2b5e3abe8cbdad6427bdf971a8abfedb2bf3a154ad05863d870400c2af61ae.jpg)  
+Figure 7: Accuracy $@ 1$ $( \% )$ of CCVR on CIFAR-10 with different numbers of virtual samples.
+
+# 5.6 Does different levels of heterogeneity affect CCVR’s performance?
+
+We study the effect of heterogeneity on CIFAR-10 by generating various non-IID partitions from Dirichlet distribution with different concentration parameters $\alpha$ . Note that partition with smaller $\alpha$ is more imbalanced. It can be seen from Table 2 that CCVR steadily improves accuracy for all the methods on all partitions. Typically, the improvements is greater when dealing with more heterogeneous data, implying that the amount of bias existing in the classifier is positively linked with the imbalanceness of training data. Another interesting discovery is that vanilla MOON performs worse than FedAvg and FedProx when $\alpha$ equals to 0.1 or 0.05, but the oracle results after classifier calibration is higher than those of FedAvg and FedProx. It indicates that MOON’s regularization on the representation brings severe negative effects on the classifier. As a consequence, MOON learns good representations but poor classifier. In that case, applying CCVR observably improves the original results, making the performance of MOON on par with FedAvg and FedProx.
+
+# 6 Limitations
+
+In this work, we mainly focus on the characteristic of the classifier in federated learning, because it is found to change the most during local training. However, our experimental results show that in a highly heterogeneous setting, only calibrating the classifier still cannot achieve comparable accuracies to that obtained on IID data. This is because the performance of classifier calibration highly relies on the quality of learned representations. Thus, it’s more important to learn a good feature space. Our experiments reveal that there may exist a trade-off in the quality of representation and classifier in federated learning on non-IID data. Namely, the methods that gain the greatest benefits from classifier calibration typically learn high-quality representations but poor classifier. We believe this finding is intriguing for future research and there is still a long way to tackling the non-IID quagmire.
+
+Moreover, we mainly focus on the image classification task in this work. Our experiments validate that the Gaussian assumption works well for visual model like CNN. However, this conclusion may not hold for language tasks or for other architectures like LSTM [46] and Transformer [47]. We believe the extensions of this work to other tasks and architectures are worth exploring.
+
+# 7 Conclusion
+
+In this work, we provide a new perspective to understand why the performance of a deep learningbased classification model degrades when trained with non-IID data in federated learning. We first anatomize the neural networks and study the similarity of different layers of the models on different clients through recent representation analysis techniques. We observe that the classifiers of different local models are less similar than any other layer, and there is a significant bias among the classifier. We then propose a novel method called Classifier Calibration with Virtual Representations (CCVR), which samples virtual features from an approximated Gaussian Mixture Model (GMM) for classifier calibration to avoid uploading raw features to the server. Experimental results on three image datasets show that CCVR steadily improves over several popular federated learning algorithms.
+
+# Acknowledgement
+
+We would like to thank the anonymous reviewers for their insightful comments and suggestions.
+
+# References
+
+[1] Deng, J., W. Dong, R. Socher, et al. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.   
+[2] McMahan, B., E. Moore, D. Ramage, et al. Communication-Efficient Learning of Deep Networks from Decentralized Data. In AISTATS, pages 1273–1282. 2017.   
+[3] Kairouz, P., H. B. McMahan, B. Avent, et al. Advances and open problems in federated learning. arXiv preprint arXiv:1912.04977, 2019.   
+[4] Li, T., A. K. Sahu, A. Talwalkar, et al. Federated learning: Challenges, methods, and future directions. IEEE Signal Processing Magazine, 37(3):50–60, 2020.   
+[5] Li, T., A. K. Sahu, M. Zaheer, et al. Federated optimization in heterogeneous networks. arXiv preprint arXiv:1812.06127, 2018.   
+[6] Zhao, Y., M. Li, L. Lai, et al. Federated learning with non-iid data. arXiv preprint arXiv:1806.00582, 2018.   
+[7] Li, Q., Y. Diao, Q. Chen, et al. Federated learning on non-iid data silos: An experimental study. arXiv preprint arXiv:2102.02079, 2021.   
+[8] Li, Q., B. He, D. Song. Model-contrastive federated learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2021.   
+[9] Karimireddy, S. P., S. Kale, M. Mohri, et al. Scaffold: Stochastic controlled averaging for federated learning. In International Conference on Machine Learning, pages 5132–5143. PMLR, 2020.   
+[10] Acar, D. A. E., Y. Zhao, R. M. Navarro, et al. Federated learning based on dynamic regularization. In International Conference on Learning Representations. 2021.   
+[11] Hsu, T.-M. H., H. Qi, M. Brown. Measuring the effects of non-identical data distribution for federated visual classification. arXiv preprint arXiv:1909.06335, 2019.   
+[12] Lin, T., L. Kong, S. U. Stich, et al. Ensemble distillation for robust model fusion in federated learning. arXiv preprint arXiv:2006.07242, 2020.   
+[13] Wang, J., Q. Liu, H. Liang, et al. Tackling the objective inconsistency problem in heterogeneous federated optimization. arXiv preprint arXiv:2007.07481, 2020.   
+[14] Yurochkin, M., M. Agarwal, S. Ghosh, et al. Bayesian nonparametric federated learning of neural networks. In International Conference on Machine Learning, pages 7252–7261. PMLR, 2019.   
+[15] Wang, H., M. Yurochkin, Y. Sun, et al. Federated learning with matched averaging. In International Conference on Learning Representations. 2020.   
+[16] Hao, W., M. El-Khamy, J. Lee, et al. Towards fair federated learning with zero-shot data augmentation. arXiv preprint arXiv:2104.13417, 2021.   
+[17] Jeong, E., S. Oh, H. Kim, et al. Communication-efficient on-device machine learning: Federated distillation and augmentation under non-iid private data. arXiv preprint arXiv:1811.11479, 2018.   
+[18] Goetz, J., A. Tewari. Federated learning via synthetic data. arXiv preprint arXiv:2008.04489, 2020.   
+[19] Fallah, A., A. Mokhtari, A. Ozdaglar. Personalized federated learning: A meta-learning approach. arXiv preprint arXiv:2002.07948, 2020.   
+[20] Jiang, Y., J. Konecnˇ y, K. Rush, et al. Improving federated learning personalization via model agnostic \` meta learning. arXiv preprint arXiv:1909.12488, 2019.   
+[21] Bui, D., K. Malik, J. Goetz, et al. Federated user representation learning. arXiv preprint arXiv:1909.12535, 2019.   
+[22] Sattler, F., K.-R. Müller, W. Samek. Clustered federated learning: Model-agnostic distributed multitask optimization under privacy constraints. IEEE Transactions on Neural Networks and Learning Systems, 2020.   
+[23] Kornblith, S., M. Norouzi, H. Lee, et al. Similarity of neural network representations revisited. In International Conference on Machine Learning, pages 3519–3529. PMLR, 2019.   
+[24] Hsieh, K., A. Phanishayee, O. Mutlu, et al. The non-iid data quagmire of decentralized machine learning. arXiv preprint arXiv:1910.00189, 2019.   
+[25] Li, X., K. Huang, W. Yang, et al. On the convergence of fedavg on non-iid data. arXiv preprint arXiv:1907.02189, 2019.   
+[26] Hsu, T.-M. H., H. Qi, M. Brown. Federated visual classification with real-world data distribution. In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part X 16, pages 76–92. Springer, 2020.   
+[27] Goodfellow, I. J., J. Pouget-Abadie, M. Mirza, et al. Generative adversarial networks. arXiv preprint arXiv:1406.2661, 2014.   
+[28] Chen, F., M. Luo, Z. Dong, et al. Federated meta-learning with fast convergence and efficient communication. arXiv preprint arXiv:1802.07876, 2018.   
+[29] Khodak, M., M.-F. F. Balcan, A. S. Talwalkar. Adaptive gradient-based meta-learning methods. In Advances in Neural Information Processing Systems, pages 5917–5928. 2019.   
+[30] Smith, V., C.-K. Chiang, M. Sanjabi, et al. Federated multi-task learning. In Advances in Neural Information Processing Systems, pages 4424–4434. 2017.   
+[31] Liang, P. P., T. Liu, L. Ziyin, et al. Think locally, act globally: Federated learning with local and global representations. arXiv preprint arXiv:2001.01523, 2020.   
+[32] Arivazhagan, M. G., V. Aggarwal, A. K. Singh, et al. Federated learning with personalization layers. arXiv preprint arXiv:1912.00818, 2019.   
+[33] Ghosh, A., J. Chung, D. Yin, et al. An efficient framework for clustered federated learning. arXiv preprint arXiv:2006.04088, 2020.   
+[34] Ghosh, A., J. Hong, D. Yin, et al. Robust federated learning in a heterogeneous environment. arXiv preprint arXiv:1906.06629, 2019.   
+[35] Xie, M., G. Long, T. Shen, et al. Multi-center federated learning. arXiv preprint arXiv:2005.01026, 2020.   
+[36] Kang, B., S. Xie, M. Rohrbach, et al. Decoupling representation and classifier for long-tailed recognition. In International Conference on Learning Representations. 2020.   
+[37] Lindsay, B. G. Mixture models: theory, geometry and applications. In NSF-CBMS regional conference series in probability and statistics, pages i–163. JSTOR, 1995.   
+[38] Vepakomma, P., T. Swedish, R. Raskar, et al. No peek: A survey of private distributed deep learning. arXiv preprint arXiv:1812.03288, 2018.   
+[39] He, C., S. Li, J. So, et al. Fedml: A research library and benchmark for federated machine learning. arXiv preprint arXiv:2007.13518, 2020.   
+[40] Krizhevsky, A., G. Hinton, et al. Learning multiple layers of features from tiny images. 2009.   
+[41] Darlow, L. N., E. J. Crowley, A. Antoniou, et al. Cinic-10 is not imagenet or cifar-10. arXiv preprint arXiv:1810.03505, 2018.   
+[42] Russakovsky, O., J. Deng, H. Su, et al. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 115(3):211–252, 2015.   
+[43] Sandler, M., A. Howard, M. Zhu, et al. Mobilenetv2: Inverted residuals and linear bottlenecks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4510–4520. 2018.   
+[44] Van der Maaten, L., G. Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(11), 2008.   
+[45] Kolouri, S., K. Nadjahi, U. Simsekli, et al. Generalized sliced wasserstein distances. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, R. Garnett, eds., Advances in Neural Information Processing Systems, vol. 32. Curran Associates, Inc., 2019.   
+[46] Hochreiter, S., J. Schmidhuber. Long short-term memory. Neural Comput., 9(8):1735–1780, 1997.   
+[47] Vaswani, A., N. Shazeer, N. Parmar, et al. Attention is all you need. In Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, page 6000–6010. Curran Associates Inc., Red Hook, NY, USA, 2017.

parse/train/AFiH_CNnVhS/AFiH_CNnVhS_content_list.json

@@ -0,0 +1,1179 @@
+[
+    {
+        "type": "text",
+        "text": "No Fear of Heterogeneity: Classifier Calibration for Federated Learning with Non-IID Data ",
+        "text_level": 1,
+        "bbox": [
+            187,
+            122,
+            812,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mi Luo1, Fei Chen2, Dapeng $\\mathbf { H } \\mathbf { u } ^ { 1 }$ , Yifan Zhang1, Jian Liang∗3, Jiashi Feng∗1 ",
+        "bbox": [
+            220,
+            224,
+            776,
+            241
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1National University of Singapore 2Huawei Noah’s Ark Lab   \n3Institute of Automation, Chinese Academy of Sciences (CAS) ",
+        "bbox": [
+            305,
+            242,
+            691,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "{romyluo7, liangjian92, jshfeng}@gmail.com chen.f@huawei.com, {dapeng.hu, yifan.zhang}@u.nus.edu ",
+        "bbox": [
+            264,
+            272,
+            732,
+            295
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            332,
+            535,
+            348
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A central challenge in training classification models in the real-world federated system is learning with non-IID data. To cope with this, most of the existing works involve enforcing regularization in local optimization or improving the model aggregation scheme at the server. Other works also share public datasets or synthesized samples to supplement the training of under-represented classes or introduce a certain level of personalization. Though effective, they lack a deep understanding of how the data heterogeneity affects each layer of a deep classification model. In this paper, we bridge this gap by performing an experimental analysis of the representations learned by different layers. Our observations are surprising: (1) there exists a greater bias in the classifier than other layers, and (2) the classification performance can be significantly improved by post-calibrating the classifier after federated training. Motivated by the above findings, we propose a novel and simple algorithm called Classifier Calibration with Virtual Representations (CCVR), which adjusts the classifier using virtual representations sampled from an approximated gaussian mixture model. Experimental results demonstrate that CCVR achieves state-of-the-art performance on popular federated learning benchmarks including CIFAR-10, CIFAR-100, and CINIC-10. We hope that our simple yet effective method can shed some light on the future research of federated learning with non-IID data. ",
+        "bbox": [
+            233,
+            361,
+            766,
+            625
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            648,
+            310,
+            665
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The rapid advances in deep learning have benefited a lot from large datasets like [1]. However, in the real world, data may be distributed on numerous mobile devices and the Internet of Things (IoT), requiring decentralized training of deep networks. Driven by such realistic needs, federated learning [2, 3, 4] has become an emerging research topic where the model training is pushed to a large number of edge clients and the raw data never leave local devices. ",
+        "bbox": [
+            174,
+            679,
+            825,
+            748
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A notorious trap in federated learning is training with non-IID data. Due to diverse user behaviors, large heterogeneity may be present in different clients’ local data, which has been found to result in unstable and slow convergence [5] and cause suboptimal or even detrimental model performance [6, 7]. There have been a plethora of works exploring promising solutions to federated learning on non-IID data. They can be roughly divided into four categories: 1) client drift mitigation [5, 8, 9, 10], which modifies the local objectives of the clients, so that the local model is consistent with the global model to a certain degree; 2) aggregation scheme [11, 12, 13, 14, 15], which improves the model fusion mechanism at the server; 3) data sharing [6, 16, 17, 18], which introduces public datasets or synthesized data to help construct a more balanced data distribution on the client or on the server; ",
+        "bbox": [
+            174,
+            755,
+            825,
+            880
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "4) personalized federated learning [19, 20, 21, 22], which aims to train personalized models for individual clients rather than a shared global model. ",
+        "bbox": [
+            173,
+            92,
+            823,
+            119
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, as suggested by [7], existing algorithms are still unable to achieve good performance on image datasets with deep learning models, and could be no better than vanilla FedAvg [2]. To identify the reasons behind this, we perform a thorough experimental investigation on each layer of a deep neural network. Specifically, we measure the Centered Kernel Alignment (CKA) [23] similarity between the representations from the same layer of different clients’ local models. The observation is thought-provoking: comparing different layers learned on different clients, the classifier has the lowest feature2 similarity across different local models. ",
+        "bbox": [
+            174,
+            126,
+            825,
+            222
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Motivated by the above discovery, we dig deeper to study the variation of the weight of the classifier in federated optimization, and confirm that the classifier tends to be biased to certain classes. After identifying this devil, we conduct several empirical trials to debias the classifier via regularizing the classifier during training or calibrating classifier weights after training. We surprisingly find that post-calibration strategy is particularly useful — with only a small fraction of IID data, the classification accuracy is significantly improved. However, this approach cannot be directly deployed in practice since it infringes the privacy rule in federated learning. ",
+        "bbox": [
+            174,
+            229,
+            825,
+            327
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Based on the above findings and considerations, we propose a novel and privacy-preserving approach called Classifier Calibration with Virtual Representations (CCVR) which rectifies the decision boundaries (the classifier) of the deep network after federated training. CCVR generates virtual representations based on an approximated Gaussian Mixture Model (GMM) in the feature space with the learned feature extractor. Experimental results show that CCVR achieves significant accuracy improvements over several popular federated learning algorithms, setting the new state-of-the-art on common federated learning benchmarks like CIFAR-10, CIFAR-100 and CINIC-10. ",
+        "bbox": [
+            174,
+            332,
+            825,
+            429
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To summarize, our contributions are threefold: (1) We present the first systematic study on the hidden representations of different layers of neural networks (NN) trained with FedAvg on non-IID data and provide a new perspective of understanding federated learning with heterogeneous data. (2) Our study reveals an intriguing fact that the primary reason for the performance degradation of NN trained on non-IID data is the classifier. (3) We propose CCVR (Classifier Calibration with Virtual Representations) — a simple and universal classifier calibration algorithm for federated learning. CCVR is built on top of the off-the-shelf feature extractor and requires no transmission of the representations of the original data, thus raising no additional privacy concern. Our empirical results show that CCVR brings considerable accuracy gains over vanilla federated learning approaches. ",
+        "bbox": [
+            174,
+            435,
+            825,
+            560
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            580,
+            321,
+            598
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Federated learning [2, 3, 4] is a fast-growing research field and remains many open problems to solve. In this work, we focus on addressing the non-IID quagmire [6, 24]. Relevant works have pursued the following four directions. ",
+        "bbox": [
+            174,
+            614,
+            825,
+            656
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Client Drift Mitigation. FedAvg [2] has been the de facto optimization method in the federated setting. However, when it is applied to the heterogeneous setting, one key issue arises: when the global model is optimized with different local objectives with local optimums far away from each other, the average of the resultant client updates (the server update) would move away from the true global optimum [9]. The cause of this inconsistency is called ‘client drift’. To alleviate it, FedAvg is compelled to use a small learning rate which may damage convergence, or reduce the number of local iterations which induces significant communication cost [25]. There have been a number of works trying to mitigate ‘client drift’ of FedAvg from various perspectives. FedProx [5] proposes to add a proximal term to the local objective which regularizes the euclidean distance between the local model and the global model. MOON [8] adopts the contrastive loss to maximize the agreement of the representation learned by the local model and that by the global model. SCAFFOLD [9] performs ‘client-variance reduction’ and corrects the drift in the local updates by introducing control variates. FedDyn [10] dynamically changes the local objectives at each communication round to ensure that the local optimum is asymptotically consistent with the stationary points of the global objective. FedIR [26] applies importance weight to the local objective, which alleviates the imbalance caused by non-identical class distributions among clients. ",
+        "bbox": [
+            174,
+            662,
+            825,
+            883
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/f73990a9b9f0a9deb71881aa7541b9f5d00d4c6813cb174aa1f1509b33573888.jpg",
+        "image_caption": [
+            "Figure 1: CKA similarities of three different layers of different ‘client model-client model’ pairs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            253,
+            90,
+            740,
+            203
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/d4cfbd236c9649235806ba08291299e2f86a36d71753e6483e99a39a2f715ec4.jpg",
+        "image_caption": [
+            "Figure 2: The means of the CKA similarities of different layers in different local models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            269,
+            222,
+            720,
+            328
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Aggregation Scheme. A fruitful avenue of explorations involves improvements at the model aggregation stage. These works are motivated by three emerging concerns. First, oscillation may occur when updating the global model using gradients collected from clients with a limited subset of labels. To alleviate it, [11] proposes FedAvgM which adopts momentum update on the server-side. Second, element-wise averaging of weights may have drastic negative effects on the performance of the averaged model. [12] shows that directly averaging local models that are learned from totally distinct data distributions cannot produce a global model that performs well on the global distribution. The authors further propose FedDF that leverages unlabeled data or artificial samples generated by GANs [27] to distill knowledge from the local models. [13] considers the setting where each client performs variable amounts of local works and proposes FedNova which normalizes the local updates before averaging. Third, a handful of works [14, 15] believe that the permutation invariance of neural network parameters may cause neuron mismatching when conducting coordinate-wise averaging of model weights. So they propose to match the parameters of local models while aggregating. ",
+        "bbox": [
+            174,
+            357,
+            825,
+            537
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Data Sharing. The key motivation behind data sharing is that a client cannot acquire samples from other clients during local training, thus the learned local model under-represents certain patterns or samples from the absent classes. The common practices are to share a public dataset [6], synthesized data [16, 17] or a condensed version of the training samples [18] to supplement training on the clients or on the server. This line of works may violate the privacy rule of federated learning since they all consider sharing raw input data of the model, either real data or artificial data. ",
+        "bbox": [
+            174,
+            542,
+            825,
+            626
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Personalized Federated Learning. Different from the above directions that aim to learn a single global model, another line of research focuses on learning personalized models. Several works aim to make the global model customized to suit the need of individual users, either by treating each client as a task in meta-learning [19, 28, 20, 29] or multi-task learning [30], or by learning both global parameters for all clients and local private parameters for individual clients [21, 31, 32]. There are also heuristic approaches that divide clients into different clusters based on their learning tasks (objectives) and perform aggregation only within the cluster [33, 34, 22, 35]. ",
+        "bbox": [
+            173,
+            632,
+            825,
+            731
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this work, we consider training a single global classification model. To the best of our knowledge, we are the first to decouple the representation and classifier in federated learning — calibrating classifier after feature learning. Strictly speaking, our proposed CCVR algorithm does not fall into any aforementioned research direction but can be readily combined with most of the existing federated learning approaches to achieve better classification performance. ",
+        "bbox": [
+            174,
+            736,
+            823,
+            805
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Heterogeneity in Federated Learning: The Devil Is in Classifier ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            825,
+            730,
+            843
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Problem Setup ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            857,
+            316,
+            872
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We aim to collaboratively train an image classification model in a federated learning system which consists of $K$ clients indexed by $[ K ]$ and a central server. Client $k$ has a local dataset $\\mathcal { D } ^ { \\bar { k } }$ , and we set $\\textstyle { \\mathcal { D } } = \\bigcup _ { k \\in [ K ] } { \\mathcal { D } } ^ { k }$ as the whole dataset. Suppose there are $C$ classes in $\\mathcal { D }$ indexed by $[ C ]$ . Denote by $( \\pmb { x } , y ) \\in \\mathcal { X } \\times [ C ]$ a sample in $\\mathcal { D }$ , where $_ { \\textbf { \\em x } }$ is an image in the input space $\\mathcal { X }$ and $y$ is its corresponding label. Let $\\mathcal { D } _ { c } ^ { k } = \\{ ( x , y ) \\in \\mathcal { D } ^ { k } : y = c \\}$ be the set of samples with ground-truth label $c$ on client $k$ . We decompose the classification model into a deep feature extractor and a linear classifier. Given a sample $( { \\pmb x } , y )$ , the feature extractor $f _ { \\pmb \\theta } : \\mathcal { X }  \\mathcal { Z }$ , parameterized by $\\pmb \\theta$ , maps the input image $_ { \\textbf { \\em x } }$ into a feature vector $z = f _ { \\pmb \\theta } ( \\pmb x ) \\in \\mathbb R ^ { d }$ in the feature space $\\mathcal { Z }$ . Then the classifier $g _ { \\varphi } : \\mathcal { Z } \\to \\mathbb { R } ^ { C }$ , parameterized by $\\varphi$ , produces a probability distribution $g _ { \\varphi } ( z )$ as the prediction for $_ { \\textbf { \\em x } }$ . Denote by ${ \\pmb w } = ( { \\pmb \\theta } , \\varphi )$ the parameter of the classification model. ",
+        "bbox": [
+            174,
+            883,
+            821,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/7d66b323c3a2a81b5e25e260d614d1f83c3e698101f31bbda35e9d4dfd4813ed.jpg",
+        "image_caption": [
+            "Figure 3: Label distribution of CIFAR-10 across clients (the first graph) and the classifier weight norm distribution across clients in different rounds and data partitions (the three graphs on the right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            88,
+            823,
+            203
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            261,
+            826,
+            380
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Federated learning proceeds through the communication between clients and the server in a roundby-round manner. In round $t$ of the process, the server sends the current model parameter $\\pmb { w } ^ { ( t - 1 ) }$ to a set $U ^ { ( t ) }$ of selected clients. Then each client $k \\in U ^ { ( t ) }$ locally updates the received parameter $\\pmb { w } ^ { ( t - 1 ) }$ to wk ${ \\pmb w } _ { k } ^ { ( t ) }$ with the following objective: ",
+        "bbox": [
+            173,
+            385,
+            825,
+            446
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a35af96232d4b8cb9cefdedc5b8a4ae3f92c1d78e4fbc3ebe274c6c3cc45833d.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\pmb { w } _ { k } ^ { ( t ) } } \\mathbb { E } _ { ( \\pmb { x } , y ) \\sim \\mathcal { D } ^ { k } } [ \\mathcal { L } ( \\pmb { w } _ { k } ^ { ( t ) } ; \\pmb { w } ^ { ( t - 1 ) } , \\pmb { x } , y ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            454,
+            627,
+            487
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L }$ is the loss function. Note that $\\mathcal { L }$ is algorithm-dependent and could rely on the current global model parameter for a number of e $w ^ { ( t - 1 ) }$ as well. For instance, FedAvg [2] computes sing the cross-entropy loss, with initialization $\\boldsymbol { w } _ { k } ^ { ( t ) }$ by running SG parameter set to $\\mathcal { D } ^ { k }$ $w ^ { ( t - 1 ) }$ FedProx [5] uses the cross entropy loss with an $L _ { 2 }$ -regularization term to constrain the distance between $w _ { k } ^ { ( t ) }$ and $\\pmb { w } ^ { ( t - 1 ) }$ ; MOON [8] introduces a contrastive loss term to address the feature drift issue. In the end of round $t$ , the selected clients send the optimized parameter back to the server and the server updates the parameter by aggregating heterogeneous parameters as follows, ",
+        "bbox": [
+            173,
+            492,
+            826,
+            602
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/95b974e6339ef54891060f2569a3b0ef7ae65e1b67f14aa06df689873ea9f489.jpg",
+        "text": "$$\n\\pmb { w } ^ { ( t ) } = \\sum _ { k \\in U ^ { ( t ) } } p _ { k } \\pmb { w } _ { k } ^ { ( t ) } , \\mathrm { ~ w h e r e ~ } p _ { k } = \\frac { | \\mathcal { D } ^ { k } | } { \\sum _ { k ^ { \\prime } \\in U ^ { ( t ) } } | \\mathcal { D } ^ { k ^ { \\prime } } | } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            607,
+            669,
+            648
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 A Closer Look at Classification Model: Classifier Bias ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            671,
+            588,
+            686
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To vividly understand how non-IID data affect the classification model in federated learning, we perform an experimental study on heterogeneous local models. For the sake of simplicity, we choose CIFAR-10 with 10 clients which is a standard federated learning benchmark, and a convolutional neural network with 7 layers used in [8]. As for the non-IID experiments, we partition the data according to the Dirichlet distribution with the concentration parameter $\\alpha$ set as 0.1. More details are covered in the Appendix. To be specific, for each layer in the model, we leverage the recently proposed Centered Kernel Alignment (CKA) [23] to measure the similarity of the output features between two local models, given the same input testing samples. CKA outputs a similarity score between 0 (not similar at all) and 1 (identical). We train the model with FedAvg for 100 communication rounds and each client optimizes for 10 local epochs at each round. ",
+        "bbox": [
+            173,
+            695,
+            825,
+            835
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We first selectively show the pairwise CKA features similarity of three different layers across local models in Figure 1. Three compared layers here are the first layer, the middle layer (Layer 4), and the last layer (the classifier), respectively. Interestingly, we find that features outputted by the deeper layer show lower CKA similarity. It indicates that, for federated models trained on non-IID data, the deeper layers have heavier heterogeneity across different clients. By averaging the pairwise CKA features similarity in Figure 1, we can obtain a single value to approximately represent the similarity of the feature outputs by each layer across different clients. We illustrate the approximated layer-wise features similarity in Figure 2. The results show that the models trained with non-IID data have consistently lower feature similarity across clients for all layers, compared with those trained on IID data. The primary finding is that, for non-IID training, the classifier shows the lowest features similarities, among all the layers. The low CKA similarities of the classifiers imply that the local classifiers change greatly to fit the local data distribution. ",
+        "bbox": [
+            174,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/8223a0b0c1ef4113b2a0ccc8974c1ab1ae2bd1cc6511e1e82e9ae4a932b807a6.jpg",
+        "table_caption": [
+            "Table 1: Accuracy $@ 1$ $( \\% )$ on CIFAR-10 with different degrees of heterogeneity. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>α= 0.5</td><td>α= 0.1</td><td>α=0.05</td></tr><tr><td>FedAvg</td><td>68.62±0.77</td><td>58.55±0.98</td><td>52.33±0.43</td></tr><tr><td>FedAvg + clsnorm</td><td>69.65±0.35 (↑ 1.03)</td><td>58.94±0.08 (↑ 0.39)</td><td>51.74±4.02 (↓ 0.59)</td></tr><tr><td>FedAvg +clsprox</td><td>68.82±0.75 (↑ 0.20)</td><td>59.04±0.70 (↑ 0.49)</td><td>52.38±0.78( (↑0.05)</td></tr><tr><td>FedAvg + clsnorm + clsprox</td><td>68.75±0.75 (↑ 0.13)</td><td>58.80±0.30 (↑ 0.25)</td><td>52.39±0.24 (↑ 0.06)</td></tr><tr><td>FedAvg + calibration (whole data)</td><td>72.51±0.53 (↑ 3.89)</td><td>64.70±0.94 (↑ 6.15)</td><td>57.53±1.00 (↑ 5.20)</td></tr></table>",
+        "bbox": [
+            202,
+            117,
+            795,
+            215
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/f06e26eac31c1eb1a9feb2f784cdfb9fee2a8da8ca825a06fb861cbae732b8b0.jpg",
+        "image_caption": [
+            "Figure 4: The effect of classifier calibration using different amounts of data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            272,
+            237,
+            720,
+            343
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            386,
+            825,
+            483
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To perform a deeper analysis on the classifier trained on non-IID data, inspired by [36], we illustrate the $L _ { 2 }$ norm of the local classifier weight vectors in Figure 3. We observe that the classifier weight norms would be biased to the class with more training samples at the initial training stage. At the end of the training, models trained on non-IID data suffer from a much heavier biased classifier than the models trained on IID data. ",
+        "bbox": [
+            174,
+            489,
+            825,
+            559
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Based on the above observations about the classifier, we hypothesize that: because the classifier is the closest layer to the local label distribution, it can be easily biased to the heterogeneous local data, reflected by the low features similarity among different local classifiers and the biased weight norms. Furthermore, we believe that debiasing the classifier is promising to directly improve the classification performance. ",
+        "bbox": [
+            173,
+            565,
+            825,
+            636
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Classifier Regularization and Calibration ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            650,
+            501,
+            666
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To effectively debias the classifier, we consider the following regularization and calibration methods. ",
+        "bbox": [
+            173,
+            676,
+            823,
+            691
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Classifier Weight $L 2$ -normalization. To eliminate the bias in classifier weight norms, we normalize the classifier weight vectors during the training and the inference stage. We abbreviate it to ‘clsnorm’. In particular, the classifier is a linear transformation with weight $\\bar { \\boldsymbol { \\varphi } } = [ \\varphi _ { 1 } , \\ldots , \\varphi _ { C } ]$ , followed by normalization and softmax. Given a feature $_ { z }$ , the output of the classifier is ",
+        "bbox": [
+            173,
+            696,
+            826,
+            752
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7486dee0a9ad0264ae83e1efa3a25e0e10d3576fc8941814104229f7d79e73d7.jpg",
+        "text": "$$\ng _ { \\varphi } ( z ) _ { i } = \\frac { e ^ { \\varphi _ { i } ^ { T } z / | | \\varphi _ { i } | | } } { \\sum _ { i ^ { \\prime } = 1 } ^ { C } e ^ { \\varphi _ { i ^ { \\prime } } ^ { T } z / | | \\varphi _ { i ^ { \\prime } } | | } } , \\quad \\forall i \\in [ C ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            758,
+            640,
+            801
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Classifier Quadratic Regularization. Beyond restricting the weight norms of classifier, we also consider adding a proximal term similar to [5] only to restrict the classifier weights to be close to the received global classifier weight vectors from the server. We write it as ‘clsprox’ for short. The loss function in Eq. (1) can be specified as ",
+        "bbox": [
+            173,
+            804,
+            825,
+            859
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/481c8a27eb96401183131bdbdd95178077b3d9a4a6c3d5eccc7d202f1c12cd22.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { w } _ { k } ^ { ( t ) } ; \\boldsymbol { w } ^ { ( t - 1 ) } , \\boldsymbol { x } , \\boldsymbol { y } ) = \\ell ( g _ { \\varphi _ { k } ^ { ( t ) } } ( f _ { \\theta _ { k } ^ { ( t ) } } ( \\boldsymbol { x } ) ) , \\boldsymbol { y } ) + \\frac { \\mu } { 2 } | | \\varphi _ { k } ^ { ( t ) } - \\varphi ^ { ( t - 1 ) } | | ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            866,
+            725,
+            893
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\ell$ is the cross-entropy loss and $\\mu$ is the regularization factor. ",
+        "bbox": [
+            174,
+            897,
+            606,
+            912
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Classifier Post-calibration with IID Samples. In addition to regularizing the classifier during federated training, we also consider a post-processing technique to adjust the learned classifier. After the federated training, we fix the feature extractor and calibrate the classifier by SGD optimization with a cross-entropy loss on IID samples. Note that this calibration strategy requires IID raw features collected from heterogeneous clients. Therefore, it can only serve as an experimental study use but cannot be applied to the real federated learning system. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            174
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments to compare the above three methods on CIFAR-10 with three different degrees of data heterogeneity and present the results in Table 1. We observe that regularizing the L2-norm of classifier weight (clsnorm) is effective for light data heterogeneity but would have less help or even lead to damages along with the increase of the heterogeneity. Regularizing the classifier parameters (clsprox) is consistently effective but with especially minor improvements. Surprisingly, we find that calibrating the classifier of the FedAvg model with all training samples brings significant performance improvement for all degrees of data heterogeneity. ",
+        "bbox": [
+            173,
+            181,
+            825,
+            279
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To further understand the classifier calibration technique, we additionally perform calibrations with different numbers of data samples and different off-the-shelf federated models trained by FedAvg and FedProx. The results are shown in Figure 4 and we observe that data-based classifier calibration performs consistently well, even with $1 / 5 0$ training data samples for calibration use. These significant performance improvements after adjusting the classifier strongly verify our aforementioned hypothesis, i.e., the devil is in the classifier. ",
+        "bbox": [
+            174,
+            284,
+            825,
+            367
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Classifier Calibration with Virtual Representations ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            388,
+            632,
+            406
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Motivated by the above observations, we propose Classifier Calibration with Virtual Representations (CCVR) that runs on the server after federated training the global model. CCVR uses virtual features drawn from an estimated Gaussian Mixture Model (GMM), without accessing any real images. Suppose $f _ { \\widehat { \\pmb { \\theta } } }$ and $g _ { \\widehat { \\varphi } }$ b bare the feature extractor and classifier of the global model, respectively, where $\\widehat { \\pmb { w } } = ( \\widehat { \\pmb { \\theta } } , \\widehat { \\pmb { \\varphi } } )$ b bis the parameter trained by a certain federated learning algorithm, e.g. FedAvg. We shall use $f _ { \\widehat { \\pmb { \\theta } } }$ to extract features and estimate the correbsponding feature distribution, and re-train $g$ using generated virtual representations. ",
+        "bbox": [
+            174,
+            421,
+            475,
+            614
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Feature Distribution Estimation. For semantics related tasks such as classification, the features learned by deep neural networks can be approximated with a mixture of Gaussian distribution. Theoretically, any continuous distribution can be approximated by using a finite number of mixture of gaussian distributions [37]. In our CCVR, we assume that features of each class in $\\mathcal { D }$ follow a Gaussian distribution. The server estimates this distribu",
+        "bbox": [
+            174,
+            621,
+            475,
+            758
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: Virtual Representation Generation ",
+        "text_level": 1,
+        "bbox": [
+            490,
+            425,
+            797,
+            440
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Input: Feature extractor $f _ { \\widehat { \\pmb { \\theta } } }$ of the global model, number $M _ { c }$ bof virtual features for class $c$ 1 # Server executes:   \n2 Send $f _ { \\widehat { \\pmb { \\theta } } }$ to clients. 3 # Clients execute: 4 foreach client $k \\in [ K ]$ do 5 foreach class $c \\in [ C ]$ do 6 Produce ${ \\ z } _ { c , k , j } = \\bar { \\ z } _ { \\widehat { \\theta } } ( \\pmb { x } _ { c , k , j } )$ for $j$ -th sample in $\\mathcal { D } _ { c } ^ { k }$ for $j \\in [ N _ { c , k } ]$ . 7 Compute $\\mu _ { c , k }$ and $\\Sigma _ { c , k }$ using Eq. (2). 8 end 9 Send $\\{ ( \\pmb { \\mu _ { c , k } } , \\pmb { \\Sigma _ { c , k } } ) : c \\in [ C ] \\}$ to server.   \n10 end   \n11 # Server executes:   \n12 foreach class $c \\in [ C ]$ do   \n13 Compute $\\pmb { \\mu } _ { c }$ and $\\Sigma _ { c }$ using Eq. (3) and (4).   \n14 Draw a set $G _ { c }$ of $M _ { c }$ features from $\\mathcal { N } ( \\mu _ { c } , \\Sigma _ { c } )$ with ground truth label $c$ .   \n15 end   \nOutput: Set of virtual representations $\\cup _ { c \\in [ C ] } G _ { c }$ ",
+        "bbox": [
+            488,
+            445,
+            816,
+            744
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "tion by computing the mean $\\pmb { \\mu } _ { c }$ and the covariance $\\Sigma _ { c }$ for each class $c$ of $\\mathcal { D }$ using gathered local statistics from clients, without accessing true data samples or their features. In particular, the server first sends the feature extractor $f _ { \\widehat { \\pmb { \\theta } } }$ of the trained global model to clients. Let $\\dot { N } _ { c , k } = | \\mathcal { D } _ { c } ^ { k } |$ be the number of samples of class $c$ θb on client $k$ , and set $\\begin{array} { r } { N _ { c } = \\sum _ { k = 1 } ^ { K } N _ { c , k } } \\end{array}$ . Client $k$ produces features $\\{ z _ { c , k , 1 } , \\dots , z _ { c , k , N _ { c , k } } \\}$ for class $c$ , where $\\boldsymbol { z } _ { c , k , j } = f _ { \\widehat { \\theta } } ( \\boldsymbol { x } _ { c , k , j } )$ is the feature of the $j$ -th sample in $\\mathcal { D } _ { c } ^ { k }$ , and computes local mean $\\mu _ { c , k }$ and covariance $\\Sigma _ { c , k }$ of $\\mathcal { D } _ { c } ^ { k }$ as: ",
+        "bbox": [
+            173,
+            760,
+            826,
+            852
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d3ffdc757beff97344c16ec554fb6f3537aa516919d1b4a475ed44f56f548326.jpg",
+        "text": "$$\n\\boldsymbol { \\mu } _ { c , k } = \\frac { 1 } { N _ { c , k } } \\sum _ { j = 1 } ^ { N _ { c , k } } z _ { c , k , j } , \\quad \\boldsymbol { \\Sigma } _ { c , k } = \\frac { 1 } { N _ { c , k } - 1 } \\sum _ { j = 1 } ^ { N _ { c , k } } \\left( z _ { c , k , j } - \\mu _ { c , k } \\right) \\left( z _ { c , k , j } - \\mu _ { c , k } \\right) ^ { T } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            863,
+            769,
+            909
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Then client $k$ uploads $\\{ ( \\pmb { \\mu _ { c , k } } , \\pmb { \\Sigma _ { c , k } } ) : c \\in [ C ] \\}$ to server. For the server to compute the global statistics of $\\mathcal { D }$ , it is sufficient to represent the global mean $\\pmb { \\mu } _ { c }$ and covariance $\\Sigma _ { c }$ using $\\mu _ { c , k }$ ’s and $\\Sigma _ { c , k }$ ’s for each class $c$ . The global mean can be straightforwardly written as ",
+        "bbox": [
+            173,
+            90,
+            825,
+            133
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0bf0f94124d02b5d35580ee45a3372fbb337ffe1a1df2fec172a1f76f8257b8f.jpg",
+        "text": "$$\n\\pmb { \\mu } _ { c } = \\frac { 1 } { N _ { c } } \\sum _ { k = 1 } ^ { K } \\sum _ { j = 1 } ^ { N _ { c , k } } z _ { c , k , j } = \\sum _ { k = 1 } ^ { K } \\frac { N _ { c , k } } { N _ { c } } \\pmb { \\mu } _ { c , k } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            143,
+            638,
+            190
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For the covariance, note that by definition we have ",
+        "bbox": [
+            173,
+            196,
+            506,
+            212
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/45e159d3ef38492e7c7fbddda8e9a4822a59ba6c4b75d652b2539982a45d477c.jpg",
+        "text": "$$\n( N _ { c , k } - 1 ) \\boldsymbol { \\Sigma } _ { c , k } = \\sum _ { j = 1 } ^ { N _ { c , k } } z _ { c , k , j } z _ { c , k , j } ^ { T } - N _ { c , k } \\cdot \\mu _ { c , k } \\mu _ { c , k } ^ { T }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            219,
+            676,
+            266
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "whenever $N _ { c , k } \\ge 1$ . Then the global covariance can be written as ",
+        "bbox": [
+            173,
+            272,
+            604,
+            287
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b1e061322630a6011de65375d726b6eb03ff382f0cf31fa732104cb5f5d97362.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\pmb { \\Sigma } _ { c } = \\frac { 1 } { N _ { c } - 1 } \\sum _ { k = 1 } ^ { K } \\sum _ { j = 1 } ^ { N _ { c , k } } z _ { c , k , j } z _ { c , k , j } ^ { T } - \\frac { N _ { c } } { N _ { c } - 1 } \\pmb { \\mu } _ { c } \\pmb { \\mu } _ { c } ^ { T } } \\\\ { \\displaystyle = \\sum _ { k = 1 } ^ { K } \\frac { N _ { c , k } - 1 } { N _ { c } - 1 } \\pmb { \\Sigma } _ { c , k } + \\sum _ { k = 1 } ^ { K } \\frac { N _ { c , k } } { N _ { c } - 1 } \\pmb { \\mu } _ { c , k } \\pmb { \\mu } _ { c , k } ^ { T } - \\frac { N _ { c } } { N _ { c } - 1 } \\pmb { \\mu } _ { c } \\pmb { \\mu } _ { c } ^ { T } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
+            297,
+            720,
+            391
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Virtual Representations Generation. After obtaining $\\pmb { \\mu } _ { c }$ ’s and $\\Sigma _ { c }$ ’s, the server generates a set $G _ { c }$ of virtual features with ground truth label $c$ from the Gaussian distribution $\\mathcal { N } ( \\mu _ { c } , \\Sigma _ { c } )$ . The number $M _ { c } : = | G _ { c } |$ of virtual features for each class $c$ could be determined by the fraction $\\frac { N _ { c } } { | \\mathcal { D } | }$ to reflect the inter-class distribution. See Algorithm 1. ",
+        "bbox": [
+            173,
+            400,
+            825,
+            462
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Classifier Re-Training. The last step of our CCVR method is classifier re-training using virtual representations. We take out the classifier $g$ from the global model, initialize its parameter as $\\widehat { \\varphi }$ , and re-train the parameter to $\\widetilde { \\varphi }$ for the objective ",
+        "bbox": [
+            173,
+            467,
+            825,
+            511
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/25716afd102726f95ad6ed7441899a172228a512d4c5415d6eb36afef3c20c91.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\tilde { \\varphi } } \\mathbb { E } _ { ( z , y ) \\sim \\bigcup _ { c \\in [ C ] } G _ { c } } [ \\ell ( g _ { \\tilde { \\varphi } } ( z ) , y ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            518,
+            612,
+            545
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\ell$ is the cross-entropy loss. We then obtain the final classification model $g _ { \\widetilde { \\varphi } } \\circ f _ { \\widehat { \\theta } }$ consisting of the pre-trained feature extractor and the calibrated classifier. ",
+        "bbox": [
+            169,
+            551,
+            825,
+            579
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Privacy Protection. CCVR protects privacy at the basic level because each client only uploads their local Gaussian statistics rather than the raw representations. Note that CCVR is just a post-hoc method, so it can be easily combined with some privacy protection techniques [38] to further secure privacy. In the Appendix, we provide an empirical analysis on the privacy-preserving aspect. ",
+        "bbox": [
+            174,
+            585,
+            825,
+            642
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Experiment ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            661,
+            303,
+            678
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Experiment Setup ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            691,
+            339,
+            707
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Federated Simulation. We consider image classification task and adopt three datasets from the popular FedML benchmark [39], i.e., CIFAR-10 [40], CIFAR-100 [40] and CINIC-10 [41]. Note that CINIC-10 is constructed from ImageNet [42] and CIFAR-10, whose samples are very similar but not drawn from identical distributions. Therefore, it naturally introduces distribution shifts which is suited to the heterogeneous nature of federated learning. To simulate federated learning scenario, we randomly split the training set of each dataset into $K$ batches, and assign one training batch to each client. Namely, each client owns its local training set. We hold out the testing set at the server for evaluation of the classification performance of the global model. For hyperparameter tuning, we first take out a $15 \\%$ subset of training set for validation. After selecting the best hyperparameter, we return the validation set to the training set and retrain the model. We are interested in the NIID partitions of the three datasets, where class proportions and number of data points of each client are unbalanced. Following [14, 15], we sample $p _ { i } \\sim D i r _ { K } ( \\alpha )$ and assign a $p _ { i , k }$ proportion of the samples from class $i$ to client $k$ . We set $\\alpha$ as 0.5 unless otherwise specified. For fair comparison, we apply the same data augmentation techniques for all methods. ",
+        "bbox": [
+            173,
+            717,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/7411b52438a19806058497ea0ca2b030964d30deea19f5b50f7ddf1d1a51c585.jpg",
+        "table_caption": [
+            "Table 2: Accuracy $@ 1$ $( \\% )$ on CIFAR-10 with different degrees of heterogeneity (α ∈ $\\{ 0 . 5 , 0 . 1 , 0 . 0 5 \\} _ { \\ r }$ ), CIFAR-100 and CINIC-10. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Method</td><td>α=0.5</td><td>α=0.1</td><td>α=0.05</td><td>CIFAR-100</td><td>CINIC-10</td></tr><tr><td rowspan=\"4\">No Calibration</td><td>FedAvg</td><td>68.62±0.77</td><td>58.55±0.98</td><td>52.33±0.43</td><td>66.25±0.54</td><td>60.20±2.04</td></tr><tr><td>FedProx</td><td>69.07±1.07</td><td>58.93±0.64</td><td>53.00±0.32</td><td>66.31±0.39</td><td>60.52±2.07</td></tr><tr><td>FedAvgM</td><td>69.00±1.68</td><td>59.22±1.14</td><td>51.98±0.91</td><td>66.43±0.23</td><td>60.46±0.73</td></tr><tr><td>MOON</td><td>70.48±0.36</td><td>57.36±0.85</td><td>49.91±0.38</td><td>67.02±0.31</td><td>65.67±2.10</td></tr><tr><td rowspan=\"4\">CCVR (Ours.)</td><td>FedAvg</td><td></td><td></td><td>71.03±0.40(↑2.41) 62.68±0.54(个4.13) 54.95±0.61(↑ 2.62)</td><td>66.60±0.63(↑0.35)</td><td>69.99±0.54 (↑9.79)</td></tr><tr><td>FedProx</td><td>70.99±1.21(个 1.92) 62.60±0.43(↑ 3.67)</td><td></td><td>55.79±1.07 (↑ 2.79) 66.61±0.48 (↑0.30)</td><td></td><td>70.05±0.66 (↑ 9.53)</td></tr><tr><td>FedAvgM</td><td>71.49±0.88 (↑ 2.49)</td><td>62.64±1.07 (个 3.42)</td><td>54.57±0.58 (↑ 2.59)</td><td>66.71±0.16(↑0.28)</td><td>70.87±0.61 (↑ 10.41)</td></tr><tr><td>MOON</td><td>71.29±0.11 (↑ 0.81)</td><td>62.22±0.70(↑ 4.86)</td><td>55.60±0.63 (↑ 5.69)</td><td>67.17±0.37 (↑ 0.15)</td><td>69.42±0.65 (↑ 3.75)</td></tr><tr><td rowspan=\"4\">Oracle</td><td>FedAvg</td><td>72.51±0.53 (↑ 3.89)</td><td>64.70±0.94(↑6.15)</td><td>57.53±1.00 (个5.20)</td><td>66.84±0.50(↑0.59)</td><td>73.47±0.30(个 13.27)</td></tr><tr><td>FedProx</td><td>72.26±1.22 (↑ 3.19)</td><td>64.63±0.93(↑ 5.70)</td><td>57.33±0.72 (↑4.33)</td><td>66.68±0.43 (↑0.37)</td><td>73.10±0.57 (↑ 12.58)</td></tr><tr><td>FedAvgM</td><td>73.30±0.19 (↑ 4.30)</td><td>64.24±1.32(↑ 5.02)</td><td>57.11±1.08 (↑ 5.13)</td><td>66.94±0.32 (↑ 0.51)</td><td>72.88±0.37 (↑ 12.42)</td></tr><tr><td>MOON</td><td>72.05±0.16 (↑ 1.57)</td><td>64.94±0.58 (个 7.58)</td><td>58.14±0.47 (个 8.23)</td><td>67.56±0.44 (↑ 0.54)</td><td>73.38±0.23 (↑ 7.71)</td></tr></table>",
+        "bbox": [
+            176,
+            131,
+            820,
+            281
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Baselines and Implementation. We consider comparing the test accuracies of the representative federated learning algorithms FedAvg [2], FedProx [5], FedAvgM [11, 26] and the state-of-the-art method MOON [8] before and after applying our CCVR. For FedProx and MOON, we carefully tune the coefficient of local regularization term $\\mu$ and report their best results. For FedAvgM, the server momentum is set to be 0.1. We use a simple 4-layer CNN network with a 2-layer MLP projection head described in [8] for CIFAR-10. For CIFAR-100 and CINIC-10, we adopt MobileNetV2 [43]. For CCVR, to make the virtual representations more Gaussian-like, we apply ReLU and Tukey’s transformation before classifier re-training. For Tukey’s transformation, the parameter is set to be 0.5. For each dataset, all methods are evaluated with the same model for fair comparison. The proposed CCVR algorithm only has one important hyperparameter, the number of feature samples $M _ { c }$ to generate. Unless otherwise stated, $M _ { c }$ is set to 100, 500 and 1000 for CIFAR-10, CIFAR100 and CINIC-10 respectively. All experiments run with PyTorch 1.7.1. More details about the implementation and datasets are summarized in the Appendix. ",
+        "bbox": [
+            173,
+            301,
+            825,
+            481
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 Can classifier calibration improve performance of federated learning? ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            497,
+            699,
+            512
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Table 2, we present the test accuracy on all datasets before and after applying our CCVR. We also report the results under an ideal setting where the whole data are available for classifier calibration (Oracle). These results indicate the upper bound of classifier calibration. ",
+        "bbox": [
+            174,
+            523,
+            825,
+            565
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "CCVR consistently improves all baseline methods. First, it can be observed that applying classifier calibration increases accuracies for all baseline methods, even with the accuracy gain up to $1 0 . 4 1 \\%$ on CINIC-10. This is particularly inspiring because CCVR requires no modification to the original federated training process. One can easily get considerable accuracy profits by simply post-processing the trained global model. Comparing the accuracy gains of different methods after applying CCVR and whole data calibration, we find that the accuracies of FedAvg and MOON get the greatest increase. On CINIC-10, the oracle results of FedAvg even outstrip those of all other baselines, implying that FedAvg focuses more on learning high-quality features but ignores learning a fair classifier. It further confirms the necessity of classifier calibration. ",
+        "bbox": [
+            174,
+            571,
+            825,
+            695
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 In what situation does CCVR work best? ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            713,
+            496,
+            727
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We observe that though there is improvement on CIFAR-100 by applying CCVR, it seems subtle compared with that of other two datasets. This is not surprising, since the final accuracy achieved by classifier calibration is not only dependent on the degree to which the classifier is debaised, but also closely correlated with the quality of pre-trained representations. In CIFAR-100, each class only has 500 training images, so the classification task itself is very difficult and the model may learn representations with low separability. It is shown that the accuracy obtained with CCVR on CIFAR-100 is very close to the upper bound, indicating that CCVR does a good job of correcting the classifier, even if it is provided with a poor feature extractor. ",
+        "bbox": [
+            174,
+            738,
+            825,
+            849
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also note that CCVR achieves huge improvements on CINIC-10. To further analyze the reason of this success and the characteristics of CCVR, we now show the t-SNE visualization [44] of the features learned by FedAvg on CINIC-10 dataset in Figure 5. From the first and second sub-graphs, we can observe that some classes dominate the classification results, while certain classes are rarely predicted correctly. For instance, the classifier makes wrong prediction for most of the samples belonging to the grey class. Another evidence showing there exists a great bias in the classifier is that, from the upper right corner of the ground truth sub-graph, we can see that the features colored green and those colored purple can be easily separated. However, due to biases in the classifier, nearly all purple features are wrongly classified as the green class. Observing the third sub-graph, we find that by applying CCVR, these misclassifications are alleviated. We also find that, with CCVR, mistakes are basically made when identifying easily-confused features that are close to the decision boundary rather than a majority of features that belong to certain classes. This suggests that the classifier weight has been adjusted to be more fair to each class. In summary, CCVR may be more effective when applied to the models with good representations but serious classifier biases. ",
+        "bbox": [
+            176,
+            856,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/f8d5b82405ebca032c0564f13f7a2ea6db7a6f88119227b3823ff0e2ce66696f.jpg",
+        "image_caption": [
+            "Figure 5: t-SNE visualization of the features learned by FedAvg on CINIC-10. The features are colored by the ground truth and the predictions of the classifier before and after applying CCVR. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
+            90,
+            750,
+            193
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            253,
+            825,
+            392
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.4 How to forecast the performance of classifier calibration? ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            411,
+            611,
+            426
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We resort to Sliced Wasserstein Distance [45], which is a popular metric to measure the distances between distributions, to quantify the separability of GMM. The experiments are conducted on CIFAR-10 with $\\alpha = 0 . 1$ . We first compute the Wasserstein distances between any two mixtures, then we average all the distances to get a mean distance. The farther the distance, the better the separability of GMM. We visualize the relationship between the accuracy gains and the separability of GMM in Figure 6. It is observed that the mean Wasserstein distance of GMM is positively correlated with the accuracy upper bound of classifier calibration. It verifies our claim in Section 5.3: CCVR may be more effective when applied to the models with good (separable) representations. In practice, one can use the mean Wasserstein distance of GMM to evaluate the quality of the simulated representations, as well as to forecast the potential performance of classifier calibration. ",
+        "bbox": [
+            174,
+            438,
+            562,
+            671
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/67b14e716f7af69fe0d9be5e2f52490fb67b213ad62a775ce51b2ba7cf1a4af5.jpg",
+        "image_caption": [
+            "Figure 6: GMM’s separability. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            580,
+            444,
+            816,
+            638
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.5 How many virtual features to generate? ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            691,
+            486,
+            705
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "One important hyperparameter in our CCVR is the number of virtual features $M _ { c }$ for each class $c$ to generate. We study the effect of $M _ { c }$ by tuning it from $\\{ 0 , 5 0 , 1 0 0 , 5 0 0 , 1 0 0 0 , 2 0 0 0 \\}$ on three different partitions of CIFAR-10 $\\alpha \\in \\{ 0 . 0 5 , 0 . 1 , 0 . 5 \\} )$ ) when applying CCVR to FedAvg. The results are provided in Figure 7. In general, even sampling only a few features can significantly increase the classification accuracy. Additionally, it is observed that on the two more heterogeneous distributions (the left two sub-graphs), more samples produces higher accuracy. Although results on NIID-0.5 give a similar hint in general, an accuracy decline when using a medium number of virtual samples is observed. This suggests that $M _ { c }$ is more sensitive when faced with a more balanced dataset. This can be explained by the nature of CCVR: utilizing virtual feature distribution to mimic the original feature distribution. As a result, if the number of virtual samples is limited, the simulated distribution may deviates from the true feature distribution. The results on NIID-0.5 implies that this trap could be easier to trigger when CCVR dealing with a more balanced original distribution. To conclude, though CCVR can provide free lunch for federated classification, one should still be very careful when tuning $M _ { c }$ to achieve higher accuracy. Generally speaking, a larger value of $M _ { c }$ is better. ",
+        "bbox": [
+            173,
+            718,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/9c2b5e3abe8cbdad6427bdf971a8abfedb2bf3a154ad05863d870400c2af61ae.jpg",
+        "image_caption": [
+            "Figure 7: Accuracy $@ 1$ $( \\% )$ of CCVR on CIFAR-10 with different numbers of virtual samples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            90,
+            802,
+            198
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5.6 Does different levels of heterogeneity affect CCVR’s performance? ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            243,
+            674,
+            258
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We study the effect of heterogeneity on CIFAR-10 by generating various non-IID partitions from Dirichlet distribution with different concentration parameters $\\alpha$ . Note that partition with smaller $\\alpha$ is more imbalanced. It can be seen from Table 2 that CCVR steadily improves accuracy for all the methods on all partitions. Typically, the improvements is greater when dealing with more heterogeneous data, implying that the amount of bias existing in the classifier is positively linked with the imbalanceness of training data. Another interesting discovery is that vanilla MOON performs worse than FedAvg and FedProx when $\\alpha$ equals to 0.1 or 0.05, but the oracle results after classifier calibration is higher than those of FedAvg and FedProx. It indicates that MOON’s regularization on the representation brings severe negative effects on the classifier. As a consequence, MOON learns good representations but poor classifier. In that case, applying CCVR observably improves the original results, making the performance of MOON on par with FedAvg and FedProx. ",
+        "bbox": [
+            173,
+            268,
+            825,
+            421
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Limitations ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            439,
+            302,
+            457
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this work, we mainly focus on the characteristic of the classifier in federated learning, because it is found to change the most during local training. However, our experimental results show that in a highly heterogeneous setting, only calibrating the classifier still cannot achieve comparable accuracies to that obtained on IID data. This is because the performance of classifier calibration highly relies on the quality of learned representations. Thus, it’s more important to learn a good feature space. Our experiments reveal that there may exist a trade-off in the quality of representation and classifier in federated learning on non-IID data. Namely, the methods that gain the greatest benefits from classifier calibration typically learn high-quality representations but poor classifier. We believe this finding is intriguing for future research and there is still a long way to tackling the non-IID quagmire. ",
+        "bbox": [
+            173,
+            470,
+            825,
+            597
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Moreover, we mainly focus on the image classification task in this work. Our experiments validate that the Gaussian assumption works well for visual model like CNN. However, this conclusion may not hold for language tasks or for other architectures like LSTM [46] and Transformer [47]. We believe the extensions of this work to other tasks and architectures are worth exploring. ",
+        "bbox": [
+            174,
+            602,
+            825,
+            657
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            678,
+            299,
+            694
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this work, we provide a new perspective to understand why the performance of a deep learningbased classification model degrades when trained with non-IID data in federated learning. We first anatomize the neural networks and study the similarity of different layers of the models on different clients through recent representation analysis techniques. We observe that the classifiers of different local models are less similar than any other layer, and there is a significant bias among the classifier. We then propose a novel method called Classifier Calibration with Virtual Representations (CCVR), which samples virtual features from an approximated Gaussian Mixture Model (GMM) for classifier calibration to avoid uploading raw features to the server. Experimental results on three image datasets show that CCVR steadily improves over several popular federated learning algorithms. ",
+        "bbox": [
+            174,
+            708,
+            825,
+            834
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgement ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            89,
+            330,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank the anonymous reviewers for their insightful comments and suggestions. ",
+        "bbox": [
+            173,
+            119,
+            802,
+            135
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            154,
+            266,
+            169
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[1] Deng, J., W. Dong, R. Socher, et al. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.   \n[2] McMahan, B., E. Moore, D. Ramage, et al. Communication-Efficient Learning of Deep Networks from Decentralized Data. In AISTATS, pages 1273–1282. 2017.   \n[3] Kairouz, P., H. B. McMahan, B. Avent, et al. Advances and open problems in federated learning. arXiv preprint arXiv:1912.04977, 2019.   \n[4] Li, T., A. K. Sahu, A. Talwalkar, et al. Federated learning: Challenges, methods, and future directions. IEEE Signal Processing Magazine, 37(3):50–60, 2020.   \n[5] Li, T., A. K. Sahu, M. Zaheer, et al. Federated optimization in heterogeneous networks. arXiv preprint arXiv:1812.06127, 2018.   \n[6] Zhao, Y., M. Li, L. Lai, et al. Federated learning with non-iid data. arXiv preprint arXiv:1806.00582, 2018.   \n[7] Li, Q., Y. Diao, Q. Chen, et al. Federated learning on non-iid data silos: An experimental study. arXiv preprint arXiv:2102.02079, 2021.   \n[8] Li, Q., B. He, D. Song. Model-contrastive federated learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2021.   \n[9] Karimireddy, S. P., S. Kale, M. Mohri, et al. Scaffold: Stochastic controlled averaging for federated learning. In International Conference on Machine Learning, pages 5132–5143. PMLR, 2020.   \n[10] Acar, D. A. E., Y. Zhao, R. M. Navarro, et al. Federated learning based on dynamic regularization. In International Conference on Learning Representations. 2021.   \n[11] Hsu, T.-M. H., H. Qi, M. Brown. Measuring the effects of non-identical data distribution for federated visual classification. arXiv preprint arXiv:1909.06335, 2019.   \n[12] Lin, T., L. Kong, S. U. Stich, et al. Ensemble distillation for robust model fusion in federated learning. arXiv preprint arXiv:2006.07242, 2020.   \n[13] Wang, J., Q. Liu, H. Liang, et al. Tackling the objective inconsistency problem in heterogeneous federated optimization. arXiv preprint arXiv:2007.07481, 2020.   \n[14] Yurochkin, M., M. Agarwal, S. Ghosh, et al. Bayesian nonparametric federated learning of neural networks. In International Conference on Machine Learning, pages 7252–7261. PMLR, 2019.   \n[15] Wang, H., M. Yurochkin, Y. Sun, et al. Federated learning with matched averaging. In International Conference on Learning Representations. 2020.   \n[16] Hao, W., M. El-Khamy, J. Lee, et al. Towards fair federated learning with zero-shot data augmentation. arXiv preprint arXiv:2104.13417, 2021.   \n[17] Jeong, E., S. Oh, H. Kim, et al. Communication-efficient on-device machine learning: Federated distillation and augmentation under non-iid private data. arXiv preprint arXiv:1811.11479, 2018.   \n[18] Goetz, J., A. Tewari. Federated learning via synthetic data. arXiv preprint arXiv:2008.04489, 2020.   \n[19] Fallah, A., A. Mokhtari, A. Ozdaglar. Personalized federated learning: A meta-learning approach. arXiv preprint arXiv:2002.07948, 2020.   \n[20] Jiang, Y., J. Konecnˇ y, K. Rush, et al. Improving federated learning personalization via model agnostic \\` meta learning. arXiv preprint arXiv:1909.12488, 2019.   \n[21] Bui, D., K. Malik, J. Goetz, et al. Federated user representation learning. arXiv preprint arXiv:1909.12535, 2019.   \n[22] Sattler, F., K.-R. Müller, W. Samek. Clustered federated learning: Model-agnostic distributed multitask optimization under privacy constraints. IEEE Transactions on Neural Networks and Learning Systems, 2020.   \n[23] Kornblith, S., M. Norouzi, H. Lee, et al. Similarity of neural network representations revisited. In International Conference on Machine Learning, pages 3519–3529. PMLR, 2019.   \n[24] Hsieh, K., A. Phanishayee, O. Mutlu, et al. The non-iid data quagmire of decentralized machine learning. arXiv preprint arXiv:1910.00189, 2019.   \n[25] Li, X., K. Huang, W. Yang, et al. On the convergence of fedavg on non-iid data. arXiv preprint arXiv:1907.02189, 2019.   \n[26] Hsu, T.-M. H., H. Qi, M. Brown. Federated visual classification with real-world data distribution. In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part X 16, pages 76–92. Springer, 2020.   \n[27] Goodfellow, I. J., J. Pouget-Abadie, M. Mirza, et al. Generative adversarial networks. arXiv preprint arXiv:1406.2661, 2014.   \n[28] Chen, F., M. Luo, Z. Dong, et al. Federated meta-learning with fast convergence and efficient communication. arXiv preprint arXiv:1802.07876, 2018.   \n[29] Khodak, M., M.-F. F. Balcan, A. S. Talwalkar. Adaptive gradient-based meta-learning methods. In Advances in Neural Information Processing Systems, pages 5917–5928. 2019.   \n[30] Smith, V., C.-K. Chiang, M. Sanjabi, et al. Federated multi-task learning. In Advances in Neural Information Processing Systems, pages 4424–4434. 2017.   \n[31] Liang, P. P., T. Liu, L. Ziyin, et al. Think locally, act globally: Federated learning with local and global representations. arXiv preprint arXiv:2001.01523, 2020.   \n[32] Arivazhagan, M. G., V. Aggarwal, A. K. Singh, et al. Federated learning with personalization layers. arXiv preprint arXiv:1912.00818, 2019.   \n[33] Ghosh, A., J. Chung, D. Yin, et al. An efficient framework for clustered federated learning. arXiv preprint arXiv:2006.04088, 2020.   \n[34] Ghosh, A., J. Hong, D. Yin, et al. Robust federated learning in a heterogeneous environment. arXiv preprint arXiv:1906.06629, 2019.   \n[35] Xie, M., G. Long, T. Shen, et al. Multi-center federated learning. arXiv preprint arXiv:2005.01026, 2020.   \n[36] Kang, B., S. Xie, M. Rohrbach, et al. Decoupling representation and classifier for long-tailed recognition. In International Conference on Learning Representations. 2020.   \n[37] Lindsay, B. G. Mixture models: theory, geometry and applications. In NSF-CBMS regional conference series in probability and statistics, pages i–163. JSTOR, 1995.   \n[38] Vepakomma, P., T. Swedish, R. Raskar, et al. No peek: A survey of private distributed deep learning. arXiv preprint arXiv:1812.03288, 2018.   \n[39] He, C., S. Li, J. So, et al. Fedml: A research library and benchmark for federated machine learning. arXiv preprint arXiv:2007.13518, 2020.   \n[40] Krizhevsky, A., G. Hinton, et al. Learning multiple layers of features from tiny images. 2009.   \n[41] Darlow, L. N., E. J. Crowley, A. Antoniou, et al. Cinic-10 is not imagenet or cifar-10. arXiv preprint arXiv:1810.03505, 2018.   \n[42] Russakovsky, O., J. Deng, H. Su, et al. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 115(3):211–252, 2015.   \n[43] Sandler, M., A. Howard, M. Zhu, et al. Mobilenetv2: Inverted residuals and linear bottlenecks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4510–4520. 2018.   \n[44] Van der Maaten, L., G. Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(11), 2008.   \n[45] Kolouri, S., K. Nadjahi, U. Simsekli, et al. Generalized sliced wasserstein distances. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, R. Garnett, eds., Advances in Neural Information Processing Systems, vol. 32. Curran Associates, Inc., 2019.   \n[46] Hochreiter, S., J. Schmidhuber. Long short-term memory. Neural Comput., 9(8):1735–1780, 1997.   \n[47] Vaswani, A., N. Shazeer, N. Parmar, et al. Attention is all you need. In Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, page 6000–6010. Curran Associates Inc., Red Hook, NY, USA, 2017. ",
+        "bbox": [
+            171,
+            167,
+            828,
+            916
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            92,
+            828,
+            915
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            92,
+            826,
+            202
+        ],
+        "page_idx": 12
+    }
+]

parse/train/H1T2hmZAb/H1T2hmZAb.md

@@ -0,0 +1,450 @@
+# DEEP COMPLEX NETWORKS
+
+Chiheb Trabelsi,∗♦♣ Olexa Bilaniuk,∗♦ Ying Zhang,† ♦♠ Dmitriy Serdyuk,† ♦ Sandeep Subramanian,† ♦
+
+João Felipe Santos,♦ Soroush Mehri,♥ Negar Rostamzadeh,♠ Yoshua Bengio♦¶ & Christopher J Pal♦♣
+
+♦ Montreal Institute for Learning Algorithms (MILA), Montreal   
+♣ Ecole Polytechnique, Montreal   
+♥ Microsoft Research, Montreal   
+♠ Element AI, Montreal   
+¶ CIFAR Senior Fellow   
+chiheb.trabelsi@polymtl.ca, olexa.bilaniuk@umontreal.ca, ying.zhlisa@gmail.com,   
+{serdyuk@iro, sandeep.subramanian.1@}umontreal.ca, jfsantos@emt.inrs.ca,   
+soroush.mehri@microsoft.com, negar@elementai.com,   
+find.me@the.web, christopher.pal@polymtl.ca
+
+# ABSTRACT
+
+At present, the vast majority of building blocks, techniques, and architectures for deep learning are based on real-valued operations and representations. However, recent work on recurrent neural networks and older fundamental theoretical analysis suggests that complex numbers could have a richer representational capacity and could also facilitate noise-robust memory retrieval mechanisms. Despite their attractive properties and potential for opening up entirely new neural architectures, complex-valued deep neural networks have been marginalized due to the absence of the building blocks required to design such models. In this work, we provide the key atomic components for complex-valued deep neural networks and apply them to convolutional feed-forward networks and convolutional LSTMs. More precisely, we rely on complex convolutions and present algorithms for complex batch-normalization, complex weight initialization strategies for complex-valued neural nets and we use them in experiments with end-to-end training schemes. We demonstrate that such complex-valued models are competitive with their realvalued counterparts. We test deep complex models on several computer vision tasks, on music transcription using the MusicNet dataset and on Speech Spectrum Prediction using the TIMIT dataset. We achieve state-of-the-art performance on these audio-related tasks.
+
+# 1 INTRODUCTION
+
+Recent research advances have made significant progress in addressing the difficulties involved in learning deep neural network architectures. Key innovations include normalization techniques (Ioffe and Szegedy, 2015; Salimans and Kingma, 2016) and the emergence of gating-based feed-forward neural networks like Highway Networks (Srivastava et al., 2015). Residual networks (He et al., 2015a; 2016) have emerged as one of the most popular and effective strategies for training very deep convolutional neural networks (CNNs). Both highway networks and residual networks facilitate the training of deep networks by providing shortcut paths for easy gradient flow to lower network layers thereby diminishing the effects of vanishing gradients (Hochreiter, 1991). He et al. (2016)
+
+show that learning explicit residuals of layers helps in avoiding the vanishing gradient problem and provides the network with an easier optimization problem. Batch normalization (Ioffe and Szegedy, 2015) demonstrates that standardizing the activations of intermediate layers in a network across a minibatch acts as a powerful regularizer as well as providing faster training and better convergence properties. Further, such techniques that standardize layer outputs become critical in deep architectures due to the vanishing and exploding gradient problems.
+
+The role of representations based on complex numbers has started to receive increased attention, due to their potential to enable easier optimization (Nitta, 2002), better generalization characteristics (Hirose and Yoshida, 2012), faster learning (Arjovsky et al., 2015; Danihelka et al., 2016; Wisdom et al., 2016) and to allow for noise-robust memory mechanisms (Danihelka et al., 2016). Wisdom et al. (2016) and Arjovsky et al. (2015) show that using complex numbers in recurrent neural networks (RNNs) allows the network to have a richer representational capacity. Danihelka et al. (2016) present an LSTM (Hochreiter and Schmidhuber, 1997) architecture augmented with associative memory with complex-valued internal representations. Their work highlights the advantages of using complex-valued representations with respect to retrieval and insertion into an associative memory. In residual networks, the output of each block is added to the output history accumulated by summation until that point. An efficient retrieval mechanism could help to extract useful information and process it within the block.
+
+In order to exploit the advantages offered by complex representations, we present a general formulation for the building components of complex-valued deep neural networks and apply it to the context of feed-forward convolutional networks and convolutional LSTMs. Our contributions in this paper are as follows:
+
+1. A formulation of complex batch normalization, which is described in Section 3.5;   
+2. Complex weight initialization, which is presented in Section 3.6;   
+3. A comparison of different complex-valued ReLU-based activation functions presented in Section 4.1;   
+4. A state of the art result on the MusicNet multi-instrument music transcription dataset, presented in Section 4.2;   
+5. A state of the art result in the Speech Spectrum Prediction task on the TIMIT dataset, presented in Section 4.3.
+
+We perform a sanity check of our deep complex network and demonstrate its effectiveness on standard image classification benchmarks, specifically, CIFAR-10, CIFAR-100. We also use a reducedtraining set of SVHN that we call $\mathrm { S V H N } ^ { * }$ . For audio-related tasks, we perform a music transcription task on the MusicNet dataset and a Speech Spectrum prediction task on TIMIT. The results obtained for vision classification tasks show that learning complex-valued representations results in performance that is competitive with the respective real-valued architectures. Our promising results in music transcription and speech spectrum prediction underscore the potential of deep complexvalued neural networks applied to acoustic related tasks1 – We continue this paper with discussion of motivation for using complex operations and related work.
+
+# 2 MOTIVATION AND RELATED WORK
+
+Using complex parameters has numerous advantages from computational, biological, and signal processing perspectives. From a computational point of view, Danihelka et al. (2016) has shown that Holographic Reduced Representations (Plate, 2003), which use complex numbers, are numerically efficient and stable in the context of information retrieval from an associative memory. Danihelka et al. (2016) insert key-value pairs in the associative memory by addition into a memory trace. Although not typically viewed as such, residual networks (He et al., 2015a; 2016) and Highway Networks (Srivastava et al., 2015) have a similar architecture to associative memories: each ResNet residual path computes a residual that is then inserted – by summing into the “memory” provided by the identity connection. Given residual networks’ resounding success on several benchmarks and their functional similarity to associative memories, it seems interesting to marry both together. This motivates us to incorporate complex weights and activations in residual networks. Together, they offer a mechanism by which useful information may be retrieved, processed and inserted in each residual block.
+
+Orthogonal weight matrices provide a novel angle of attack on the well-known vanishing and exploding gradient problems in RNNs. Unitary RNNs (Arjovsky et al., 2015) are based on unitary weight matrices, which are a complex generalization of orthogonal weight matrices. Compared to their orthogonal counterparts, unitary matrices provide a richer representation, for instance being capable of implementing the discrete Fourier transform, and thus of discovering spectral representations. Arjovsky et al. (2015) show the potential of this type of recurrent neural networks on toy tasks. Wisdom et al. (2016) provided a more general framework for learning unitary matrices and they applied their method on toy tasks and on a real-world speech task.
+
+Using complex weights in neural networks also has biological motivation. Reichert and Serre (2013) have proposed a biologically plausible deep network that allows one to construct richer and more versatile representations using complex-valued neuronal units. The complex-valued formulation allows one to express the neuron’s output in terms of its firing rate and the relative timing of its activity. The amplitude of the complex neuron represents the former and its phase the latter. Input neurons that have similar phases are called synchronous as they add constructively, whereas asynchronous neurons add destructively and thus interfere with each other. This is related to the gating mechanism used in both deep feed-forward neural networks (Srivastava et al., 2015; van den Oord et al., 2016a;b) and recurrent neural networks (Hochreiter and Schmidhuber, 1997; Cho et al., 2014; Zilly et al., 2016) as this mechanism learns to synchronize inputs that the network propagates at a given feed-forward layer or time step. In the context of deep gating-based networks, synchronization means the propagation of inputs whose controlling gates simultaneously hold high values. These controlling gates are usually the activations of a sigmoid function. This ability to take into account phase information might explain the effectiveness of incorporating complex-valued representations in the context of recurrent neural networks.
+
+The phase component is not only important from a biological point of view but also from a signal processing perspective. It has been shown that the phase information in speech signals affects their intelligibility (Shi et al., 2006). Also Oppenheim and Lim (1981) show that the amount of information present in the phase of an image is sufficient to recover the majority of the information encoded in its magnitude. In fact, phase provides a detailed description of objects as it encodes shapes, edges, and orientations.
+
+Recently, Rippel et al. (2015) leveraged the Fourier spectral representation for convolutional neural networks, providing a technique for parameterizing convolution kernel weights in the spectral domain, and performing pooling on the spectral representation of the signal. However, the authors avoid performing complex-valued convolutions, instead building from real-valued kernels in the spatial domain. In order to ensure that a complex parametrization in the spectral domain maps onto real-valued kernels, the authors impose a conjugate symmetry constraint on the spectral-domain weights, such that when the inverse Fourier transform is applied to them, it only yields real-valued kernels.
+
+As pointed out in Reichert and Serre (2013), the use of complex-valued neural networks (Georgiou and Koutsougeras, 1992; Zemel et al., 1995; Kim and Adalı, 2003; Hirose, 2003; Nitta, 2004) has been investigated long before the earliest deep learning breakthroughs (Hinton et al., 2006; Bengio et al., 2007; Poultney et al., 2007). Recently Reichert and Serre (2013); Bruna et al. (2015); Arjovsky et al. (2015); Danihelka et al. (2016); Wisdom et al. (2016) have tried to bring more attention to the usefulness of deep complex neural networks by providing theoretical and mathematical motivation for using complex-valued deep networks. However, to the best of our knowledge, most of the recent works using complex valued networks have been applied on toy tasks, with the exception of some attempts. In fact, (Oyallon and Mallat, 2015; Tygert et al., 2015; Worrall et al., 2016) have used complex representation in vision tasks. Wisdom et al. (2016) have also performed a real-world speech task consisting of predicting the log magnitude of the future short time Fourier transform frames. In Natural Language Processing, (Trouillon et al., 2016; Trouillon and Nickel, 2017) have used complex-valued embeddings. Much remains to be done to develop proper tools and a general framework for training deep neural networks with complex-valued parameters.
+
+Given the compelling reasons for using complex-valued representations, the absence of such frameworks represents a gap in machine learning tooling, which we fill by providing a set of building blocks for deep complex-valued neural networks that enable them to achieve competitive results with their real-valued counterparts on real-world tasks.
+
+# 3 COMPLEX BUILDING BLOCKS
+
+In this section, we present the core of our work, laying down the mathematical framework for implementing complex-valued building blocks of a deep neural network.
+
+# 3.1 REPRESENTATION OF COMPLEX NUMBERS
+
+We start by outlining the way in which complex numbers are represented in our framework. A complex number $z = a + i b$ has a real component $a$ and an imaginary component $b$ . We represent the real part $a$ and the imaginary part $b$ of a complex number as logically distinct real valued entities and simulate complex arithmetic using real-valued arithmetic internally. Consider a typical realvalued $2 D$ convolution layer that has $N$ feature maps such that $N$ is divisible by 2; to represent these as complex numbers, we allocate the first $N / 2$ feature maps to represent the real components and the remaining $N / 2$ to represent the imaginary ones. Thus, for a four dimensional weight tensor $W$ that links $N _ { i n }$ input feature maps to $N _ { o u t }$ output feature maps and whose kernel size is $m \times m$ we would have a weight tensor of size $\left( N _ { o u t } \times N _ { i n } \times m \times m \right) / 2$ complex weights.
+
+# 3.2 COMPLEX CONVOLUTION
+
+In order to perform the equivalent of a traditional real-valued 2D convolution in the complex domain, we convolve a complex filter matrix $\mathbf { W } = \mathbf { A } + i \mathbf { B }$ by a complex vector $\mathbf { h } = \mathbf { x } + i \mathbf { y }$ where $\mathbf { A }$ and $\mathbf { B }$ are real matrices and $\mathbf { X }$ and $\mathbf { y }$ are real vectors since we are simulating complex arithmetic using real-valued entities. As the convolution operator is distributive, convolving the vector $\mathbf { h }$ by the filter W we obtain:
+
+$$
+\mathbf { W } * \mathbf { h } = \mathbf { \Gamma } \bigl ( \mathbf { A } * \mathbf { x } - \mathbf { B } * \mathbf { y } \bigr ) + i \bigl ( \mathbf { B } * \mathbf { x } + \mathbf { A } * \mathbf { y } \bigr ) .
+$$
+
+As illustrated in Figure 1a, if we use matrix notation to represent real and imaginary parts of the convolution operation we have:
+
+$$
+\left[ \Re ( \mathbf { W } * \mathbf { h } ) \right] = \left[ \mathbf { A } - \mathbf { B } \right] * \left[ \mathbf { x } \right] .
+$$
+
+# 3.3 COMPLEX DIFFERENTIABILITY
+
+In order to perform backpropagation in a complex-valued neural network, a sufficient condition is to have a cost function and activations that are differentiable with respect to the real and imaginary parts of each complex parameter in the network. See Section 6.3 in the Appendix for the complex chain rule.
+
+By constraining activation functions to be complex differentiable or holomorphic, we restrict the use of possible activation functions for a complex valued neural networks (For further details about holomorphism please refer to Section 6.2 in the appendix). Hirose and Yoshida (2012) shows that it is unnecessarily restrictive to limit oneself only to holomorphic activation functions; Those functions that are differentiable with respect to the real part and the imaginary part of each parameter are also compatible with backpropagation. (Arjovsky et al., 2015; Wisdom et al., 2016; Danihelka et al., 2016) have used non-holomorphic activation functions and optimized the network using regular, real-valued backpropagation to compute partial derivatives of the cost with respect to the real and imaginary parts.
+
+Even though their use greatly restricts the set of potential activations, it is worth mentioning that holomorphic functions can be leveraged for computational efficiency purposes. As pointed out in Sarroff et al. (2015), using holomorphic functions allows one to share gradient values (because the activation satisfies the Cauchy-Riemann equations 11 and 12 in the appendix). So, instead of computing and backpropagating 4 different gradients, only 2 are required.
+
+# 3.4.1 MODRELU
+
+Numerous activation functions have been proposed in the literature in order to deal with complexvalued representations. (Arjovsky et al., 2015) have proposed modReLU, which is defined as follows:
+
+$$
+\begin{array} { r } { \mathrm { \ m o d R e L U } ( z ) = \mathrm { R e L U } ( | z | + b ) e ^ { i \theta _ { z } } = \left\{ \begin{array} { l l } { \left( | z | + b \right) \frac { z } { | z | } } & { \mathrm { i f ~ } | z | + b \ge 0 , } \\ { 0 } & { \mathrm { o t h e r w i s e } , } \end{array} \right. } \end{array}
+$$
+
+where $z \in \mathbb { C }$ , $\theta _ { z }$ is the phase of $z$ , and $b \in \mathbb { R }$ is a learnable parameter. As $| z |$ is always positive, a bias $b$ is introduced in order to create a “dead zone” of radius $b$ around the origin 0 where the neuron is inactive, and outside of which it is active. The authors have used modReLU in the context of unitary RNNs. Their design of modReLU is motivated by the fact that applying separate ReLUs on both real and imaginary parts of a neuron performs poorly on toy tasks. The intuition behind the design of modReLU is to preserve the pre-activated phase $\theta _ { z }$ , as altering it with an activation function severely impacts the complex-valued representation. modReLU does not satisfy the Cauchy-Riemann equations, and thus is not holomorphic. We have tested modReLU in deep feed-forward complex networks and the results are given in Table 6.4.
+
+# 3.4.2 CRELU AND zRELU
+
+We call Complex ReLU (or CReLU) the complex activation that applies separate ReLUs on both of the real and the imaginary part of a neuron, i.e:
+
+$$
+\begin{array} { r } { \mathbb { C } \mathrm { R e } \mathrm { L U } ( z ) = \mathrm { R e } \mathrm { L U } ( \mathfrak { R } ( z ) ) + i \mathrm { R e } \mathrm { L U } ( \mathfrak { I } ( z ) ) . } \end{array}
+$$
+
+CReLU satisfies the Cauchy-Riemann equations when both the real and imaginary parts are at the same time either strictly positive or strictly negative. This means that CReLU satisfies the CauchyRiemann equations when $\theta _ { z } \in ] 0 , \pi / 2 [$ or $\overline { { \theta _ { z } } } \overline { { \angle } } \overline { { \angle } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } } \overline { { \vert } { \vert } }$ [. We have tested CReLU in deep feedforward neural networks and the results are given in Table 6.4.
+
+It is also worthwhile to mention the work done by Guberman (2016) where a ReLU-based complex activation which satisfies the Cauchy-Riemann equations everywhere except for the set of points $\{ \Re ( z ) > 0 , \Im ( z ) = 0 \} \cup \{ \Re ( z ) = 0 , \Im ( z ) > 0 \}$ ias used. The activation function has similarities to CReLU. We call Guberman (2016) activation as $z$ ReLU and is defined as follows:
+
+$$
+z \mathrm { R e L U } ( z ) = \left\{ z \begin{array} { l l } { z } & { \mathrm { i f } \theta _ { z } \in [ 0 , \pi / 2 ] , } \\ { 0 } & { \mathrm { o t h e r w i s e } , } \end{array} \right.
+$$
+
+We have tested $z$ ReLU in deep feed-forward complex networks and the results are given in Table 6.4.
+
+# 3.5 COMPLEX BATCH NORMALIZATION
+
+Deep networks generally rely upon Batch Normalization (Ioffe and Szegedy, 2015) to accelerate learning. In some cases batch normalization is essential to optimize the model. The standard formulation of Batch Normalization applies only to real values. In this section, we propose a batch normalization formulation that can be applied for complex values.
+
+To standardize an array of complex numbers to the standard normal complex distribution, it is not sufficient to translate and scale them such that their mean is 0 and their variance 1. This type of normalization does not ensure equal variance in both the real and imaginary components, and the resulting distribution is not guaranteed to be circular; It will be elliptical, potentially with high eccentricity.
+
+We instead choose to treat this problem as one of whitening 2D vectors, which implies scaling the data by the square root of their variances along each of the two principal components. This can be done by multiplying the 0-centered data $( { \pmb x } - { \mathbb E } [ { \pmb x } ] )$ by the inverse square root of the $2 \times 2$ covariance matrix $V$ :
+
+$$
+\tilde { \pmb { x } } = ( \pmb { V } ) ^ { - \frac 1 2 } \left( \pmb { x } - \mathbb { E } [ \pmb { x } ] \right) ,
+$$
+
+where the covariance matrix $V$ is
+
+$$
+\begin{array} { r } { V = \left( \begin{array} { l l } { V _ { r r } } & { V _ { r i } } \\ { V _ { i r } } & { V _ { i i } } \end{array} \right) = \left( \begin{array} { l l } { \mathrm { C o v } ( \Re \{ x \} , \Re \{ x \} ) } & { \mathrm { C o v } ( \Re \{ x \} , \Im \{ x \} ) } \\ { \mathrm { C o v } ( \Im \{ x \} , \Re \{ x \} ) } & { \mathrm { C o v } ( \Im \{ x \} , \Im \{ x \} ) } \end{array} \right) . } \end{array}
+$$
+
+The square root and inverse of $2 \times 2$ matrices has an inexpensive, analytical solution, and its existence is guaranteed by the positive (semi-)definiteness of $V$ . Positive definiteness of $V$ is ensured by the addition of $\epsilon I$ to $V$ (Tikhonov regularization). The mean subtraction and multiplication by the inverse square root of the variance ensures that $\tilde { \pmb x }$ has standard complex distribution with mean $\mu = 0$ , covariance $\Gamma = 1$ and pseudo-covariance (also called relation) $C = 0$ . The mean, the covariance and the pseudo-covariance are given by:
+
+$$
+\begin{array} { r l } & { \mu = \operatorname { \mathbb { E } } \left[ \tilde { { \boldsymbol { x } } } \right] } \\ & { \Gamma = \operatorname { \mathbb { E } } \left[ \left( \tilde { { \boldsymbol { x } } } - { \boldsymbol { \mu } } \right) ( \tilde { { \boldsymbol { x } } } - { \boldsymbol { \mu } } ) ^ { * } \right] = V _ { r r } + V _ { i i } + i \left( V _ { i r } - V _ { r i } \right) } \\ & { C = \operatorname { \mathbb { E } } \left[ \left( \tilde { { \boldsymbol { x } } } - { \boldsymbol { \mu } } \right) ( \tilde { { \boldsymbol { x } } } - { \boldsymbol { \mu } } ) \right] = V _ { r r } - V _ { i i } + i \left( V _ { i r } + V _ { r i } \right) . } \end{array}
+$$
+
+The normalization procedure allows one to decorrelate the imaginary and real parts of a unit. This has the advantage of avoiding co-adaptation between the two components which reduces the risk of overfitting (Cogswell et al., 2015; Srivastava et al., 2014).
+
+Analogously to the real-valued batch normalization algorithm, we use two parameters, $\beta$ and $\gamma$ . The shift parameter $\beta$ is a complex parameter with two learnable components (the real and imaginary means). The scaling parameter $\gamma$ is a $2 \times 2$ positive semi-definite matrix with only three degrees of freedom, and thus only three learnable components. In much the same way that the matrix $( V ) ^ { - { \frac { 1 } { 2 } } }$ normalized the variance of the input to 1 along both of its original principal components, so does $\gamma$ scale the input along desired new principal components to achieve a desired variance. The scaling parameter $\gamma$ is given by:
+
+$$
+\gamma = \left( \begin{array} { c c } { { \gamma _ { r r } } } & { { \gamma _ { r i } } } \\ { { \gamma _ { r i } } } & { { \gamma _ { i i } } } \end{array} \right) .
+$$
+
+As the normalized input $\tilde { \pmb x }$ has real and imaginary variance 1, we initialize both $\gamma _ { r r }$ and $\gamma _ { i i }$ to $1 / \sqrt { 2 }$ in order to obtain a modulus of 1 for the variance of the normalized value. $\gamma _ { r i }$ , $\Re \{ \beta \}$ and $\Im \{ \beta \}$ are initialized to 0. The complex batch normalization is defined as:
+
+$$
+\mathrm { B N } \left( \tilde { \mathbf { x } } \right) = \gamma \tilde { \mathbf { x } } + \beta .
+$$
+
+We use running averages with momentum to maintain an estimate of the complex batch normalization statistics during training and testing. The moving averages of $V _ { r i }$ and $\beta$ are initialized to 0. The moving averages of $V _ { r r }$ and $V _ { i i }$ are initialized to $1 / \bar { \sqrt { 2 } }$ . The momentum for the moving averages is set to 0.9.
+
+# 3.6 COMPLEX WEIGHT INITIALIZATION
+
+In a general case, particularly when batch normalization is not performed, proper initialization is critical in reducing the risks of vanishing or exploding gradients. To do this, we follow the same steps as in Glorot and Bengio (2010) and He et al. (2015b) to derive the variance of the complex weight parameters.
+
+A complex weight has a polar form as well as a rectangular form
+
+$$
+W = | W | e ^ { i \theta } = \Re \{ W \} + i \Im \{ W \} ,
+$$
+
+where $\theta$ and $| W |$ are respectively the argument (phase) and magnitude of $W$ .
+
+Variance is the difference between the expectation of the squared magnitude and the square of the expectation:
+
+$$
+\operatorname { V a r } ( W ) = \operatorname { \mathbb { E } } \left[ W W ^ { * } \right] - ( \operatorname { \mathbb { E } } \left[ W \right] ) ^ { 2 } = \operatorname { \mathbb { E } } \left[ | W | ^ { 2 } \right] - ( \operatorname { \mathbb { E } } \left[ W \right] ) ^ { 2 } ,
+$$
+
+which reduces, in the case of $W$ symmetrically distributed around 0, to $\mathbb { E } \left[ | W | ^ { 2 } \right]$ . We do not know yet the value of $\operatorname { V a r } ( W ) = \mathbb { E } \left[ | W | ^ { 2 } \right]$ . However, we do know a related quantity, $\mathrm { V a r } ( | W | )$ , because the magnitude of complex normal values, $| W |$ , follows the Rayleigh distribution (Chi-distributed with two degrees of freedom (DOFs)). This quantity is
+
+$$
+\operatorname { V a r } ( | W | ) = \operatorname { \mathbb { E } } \left[ | W | | W | ^ { * } \right] - ( \operatorname { \mathbb { E } } \left[ | W | \right] ) ^ { 2 } = \operatorname { \mathbb { E } } \left[ | W | ^ { 2 } \right] - ( \operatorname { \mathbb { E } } \left[ | W | \right] ) ^ { 2 } .
+$$
+
+Putting them together:
+
+$\operatorname { V a r } ( | W | ) = \operatorname { V a r } ( W ) - ( \mathbb { E } \left[ | W | \right] ) ^ { 2 }$ , and $\operatorname { V a r } ( W ) = \operatorname { V a r } ( | W | ) + ( \mathbb { E } \left[ | W | \right] ) ^ { 2 } .$
+
+We now have a formulation for the variance of $W$ in terms of the variance and expectation of its magnitude, both properties analytically computable from the Rayleigh distribution’s single parameter, $\sigma$ , indicating the mode. These are:
+
+$$
+\mathbb { E } \left[ \lvert W \rvert \right] = \sigma \sqrt { \frac { \pi } { 2 } } , ~ \mathrm { V a r } ( \lvert W \rvert ) = \frac { 4 - \pi } { 2 } \sigma ^ { 2 } .
+$$
+
+The variance of $W$ can thus be expressed in terms of its generating Rayleigh distribution’s single parameter, $\sigma$ , thus:
+
+$$
+\operatorname { V a r } ( W ) = { \frac { 4 - \pi } { 2 } } \sigma ^ { 2 } + \left( \sigma { \sqrt { \frac { \pi } { 2 } } } \right) ^ { 2 } = 2 \sigma ^ { 2 } .
+$$
+
+If we want to respect the Glorot and Bengio (2010) criterion which ensures that the variances of the input, the output and their gradients are the same, then we would have $\operatorname { V a r } ( W ) = 2 / ( n _ { i n } +$ $n _ { o u t } )$ ), where √ $n _ { i n }$ and $n _ { o u t }$ are the number of input and output units respectively. In such case, $\sigma = 1 / \sqrt { n _ { i n } + n _ { o u t } }$ . If we want to respect the He et al. (2015b) initialization that presents an initialization criterion that is specific to ReLUs, then $\operatorname { V a r } ( W ) = 2 / n _ { i n }$ which $\sigma = 1 / \sqrt { n _ { i n } }$ .
+
+The magnitude of the complex parameter $W$ is then initialized using the Rayleigh distribution with the appropriate mode $\sigma$ . We can see from equation 10, that the variance of $W$ depends on on its magnitude and not on its phase. We then initialize the phase using the uniform distribution between $- \pi$ and $\pi$ . By performing the multiplication of the magnitude by the phasor as is detailed in equation 8, we perform the complete initialization of the complex parameter.
+
+In all the experiments that we report, we use variant of this initialization which leverages the independence property of unitary matrices. As it is stated in Cogswell et al. (2015), Srivastava et al. (2014), and Tompson et al. (2015), learning decorrelated features is beneficial for learning as it allows to perform better generalization and faster learning. This motivates us to achieve initialization by considering a (semi-)unitary matrix which is reshaped to the size of the weight tensor. Once this is done, the weight tensor is mutiplied by $\sqrt { H e _ { v a r } / \mathrm { V a r } ( W ) }$ or $\sqrt { G l o r o t _ { v a r } / \mathrm { V a r } ( W ) }$ where $G l o r o t _ { v a r }$ and $H e _ { v a r }$ are respectively equal to $2 / ( n _ { i n } + n _ { o u t } )$ and $2 / n _ { i n }$ . In such a way we allow kernels to be independent from each other as much as possible while respecting the desired criterion. Note that we perform the analogous initialization for real-valued models by leveraging the independence property of orthogonal matrices in order to build kernels that are as much independent from each other as possible while respecting a given criterion.
+
+# 3.7 COMPLEX CONVOLUTIONAL RESIDUAL NETWORK
+
+A deep convolutional residual network of the nature presented in He et al. (2015a; 2016) consists of 3 stages within which feature maps maintain the same shape. At the end of a stage, the feature maps are downsampled by a factor of 2 and the number of convolution filters are doubled. The sizes of the convolution kernels are always set to $3 \mathrm { ~ x ~ } 3$ . Within a stage, there are several residual blocks which comprise 2 convolution layers each. The contents of one such residual block in the real and complex setting is illustrated in Appendix Figure 1b.
+
+In the complex valued setting, the majority of the architecture remains identical to the one presented in He et al. (2016) with a few subtle differences. Since all datasets that we work with have realvalued inputs, we present a way to learn their imaginary components to let the rest of the network operate in the complex plane. We learn the initial imaginary component of our input by performing the operations present within a single real-valued residual block
+
+$$
+B N  R e L U  C o n v  B N  R e L U  C o n v
+$$
+
+Using this learning block yielded better emprical results than assuming that the input image has a null imaginary part. The parameters of this real-valued residual block are trained by backpropagating errors from the task specific loss function. Secondly, we perform a $C o n v  \dot { B N }  \mathsf { \bar { A } } c \dot { t } i \dot { v } a t i o n$ operation on the obtained complex input before feeding it to the first residual block. We also perform the same operation on the real-valued network input instead of $C o n v  M$ axpooling as in He et al. (2016). Inside, residual blocks, we subtly alter the way in which we perform a projection at
+
+Table 1: Classification error on CIFAR-10, CIFAR-100 and $\mathrm { S V H N ^ { * } }$ using different complex activations functions (zReLU, modReLU and CReLU). WS, DN and IB stand for the wide and shallow, deep and narrow and in-between models respectively. The prefixes R & C refer to the real and complex valued networks respectively. Performance differences between the real network and the complex network using CReLU are reported between their respective best models. All models are constructed to have roughly 1.7M parameters except the modReLU models which have roughly $2 . 5 \mathbf { M }$ parameters. modReLU and zReLU were largely outperformed by CReLU in the reported experiments. Due to limited resources, we haven’t performed all possible experiments as the conducted ones are already conclusive. A "-" is filled in front of an unperformed experiment.
+
+Table 2: Classification error on CIFAR-10, CIFAR-100 and $\operatorname { S V H N } ^ { * }$ using different normalization strategies. NCBN, CBN and BN stand for a Naive variant of the complex batch-normalization, complex batch-normalization and regular batch normalization respectively. (R) & (C) refer to the use of the real- and complex-valued convolution respectively. The complex models use CReLU as activation. All models are constructed to have roughly 1.7M parameters. 5 out of 6 experiments using the naive variant of the complex batch normalization failed with the apparition of NaNs during training. As these experiments are already conclusive and due to limited resources, we haven’t conducted other experiments for the NCBN model. A "-" is filled in front of an unperformed experiment.   
+
+<table><tr><td rowspan=1 colspan=1>ARCH</td><td rowspan=1 colspan=3>CIFAR-10</td><td rowspan=1 colspan=1>CIFAR-100</td><td rowspan=1 colspan=1>SVHN*</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=4>zRELU MODRELU CRELU zRELU MODRELU (CRELU</td><td rowspan=1 colspan=1>zRELU MODRELU CRELU</td></tr><tr><td rowspan=2 colspan=1>cwsCDN</td><td rowspan=1 colspan=3>11.71   23.42   6.17</td><td rowspan=1 colspan=1>50.38   26.36</td><td rowspan=1 colspan=1>80.41   7.43    3.70</td></tr><tr><td rowspan=1 colspan=1>9.</td><td rowspan=1 colspan=2>22.49   6.73</td><td rowspan=2 colspan=1>50.64   28.2248.10   28.64</td><td rowspan=1 colspan=1>80.41      1      3.72</td></tr><tr><td rowspan=1 colspan=1>CIB</td><td rowspan=1 colspan=3>11.36   23.63   5.59</td><td rowspan=1 colspan=1>4.98      1      3.62</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>RELU</td><td rowspan=1 colspan=1>RELU</td><td rowspan=1 colspan=1>RELU</td></tr><tr><td rowspan=1 colspan=1>RWS</td><td rowspan=1 colspan=3>5.42</td><td rowspan=2 colspan=1>27.2227.8427.71</td><td rowspan=2 colspan=1>3.423.524.30</td></tr><tr><td rowspan=1 colspan=1>RDNRIB</td><td rowspan=1 colspan=3>6.296.07</td><td rowspan=1 colspan=1>6.29</td><td rowspan=1 colspan=1>3.52</td></tr><tr><td rowspan=1 colspan=1>DIFF</td><td rowspan=1 colspan=3>-0.17</td><td rowspan=1 colspan=1>+0.86</td><td rowspan=1 colspan=1>-0.20</td></tr></table>
+
+<table><tr><td>ARCH</td><td colspan="3">CIFAR-10</td><td colspan="3">CIFAR-100</td><td colspan="3">SVHN*</td></tr><tr><td></td><td>NCBN(C)</td><td>CBN(R)</td><td>BN(C)</td><td>NCBN(C)</td><td>CBN(R)</td><td>BN(C)</td><td>NCBN(C)</td><td>CBN(R)</td><td>BN(C)</td></tr><tr><td>WS</td><td>1</td><td>5.47</td><td>6.32</td><td>27.29</td><td>26.63</td><td>27.89</td><td>NAN</td><td>3.80</td><td>3.52</td></tr><tr><td>DN</td><td></td><td>5.89</td><td>6.71</td><td>NAN</td><td>27.13</td><td>28.83</td><td>NAN</td><td>3.54</td><td>3.58</td></tr><tr><td>IB</td><td>-</td><td>5.66</td><td>6.83</td><td>NAN</td><td>26.99</td><td>29.89</td><td>NAN</td><td>3.74</td><td>3.56</td></tr></table>
+
+the end of a stage in our network. We concatenate the output of the last residual block with the output of a 1x1 convolution applied on it with the same number of filters used throughout the stage and subsample by a factor of 2. In contrast, He et al. (2016) perform a similar 1x1 convolution with twice the number of feature filters in the current stage to both downsample the feature maps spatially and double them in number.
+
+# 4 EXPERIMENTAL RESULTS
+
+In this section, we present empirical results from using our model to perform image, music classification and spectrum prediction. First, we present our model’s architecture followed by the results we obtained on CIFAR-10, CIFAR-100, and $\mathrm { S V H N ^ { * } }$ as well as the results on automatic music transcription on the MusicNet benchmark and speech spectrum prediction on TIMIT.
+
+# 4.1 IMAGE RECOGNITION
+
+We adopt an architecture inspired by He et al. (2016). The latter will also serve as a baseline to compare against. We train comparable real-valued Neural Networks using the standard ReLU activation function. We have tested our complex models with the CReLU, zReLU and modRelu activation functions. We use a cross entropy loss for both real and complex models. A global average pooling layer followed by a single fully connected layer with a softmax function is used to classify the input as belonging to one of 10 classes in the CIFAR-10 and SVHN datasets and 100 classes for CIFAR-100.
+
+We consider architectures that trade-off model depth (number of residual blocks per stage) and width (number of convolutional filters in each layer) given a fixed parameter budget. Specifically, we build three different models - wide and shallow (WS), deep and narrow (DN) and in-between (IB). In a model that has roughly 1.7 million parameters, our WS architecture for a complex network starts with 12 complex filters (24 real filters) per convolution layer in the initial stage and 16 residual blocks per stage. The DN architecture starts with 10 complex filters and 23 blocks per stage while the IB variant starts with 11 complex filters and 19 blocks per stage. The real-valued counterpart has also 1.7 million parameters. Its WS architecture starts with 18 real filters per convolutional layer and 14 blocks per stage. The DN architecture starts with 14 real filters and 23 blocks per stage and the IB architecture starts with 16 real filters and 18 blocks per stage.
+
+All models (real and complex) were trained using the backpropagation algorithm with Stochastic Gradient Descent with Nesterov momentum (Nesterov, 1983) set at 0.9. We also clip the norm of our gradients to 1. We tweaked the learning rate schedule used in He et al. (2016) in both the real and complex residual networks to extract small performance improvements in both. We start our learning rate at 0.01 for the first 10 epochs to warm up the training and then set it at 0.1 from epoch 10-100 and then anneal the learning rates by a factor of 10 at epochs 120 and 150. We end the training at epoch 200.
+
+Table 6.4 presents our results on performing image classification on CIFAR-10, CIFAR-100. In addition, we also consider a truncated version of the Street View House Numbers (SVHN) dataset which we call $\operatorname { S V H N } ^ { * }$ . For computational reasons, we use the required 73,257 training images of Street View House Numbers (SVHN). We still test on all 26,032 images. For all the tasks and for both the real- and complex-valued models, The WS architecture has yielded the best performances. This is in concordance with Zagoruyko and Komodakis (2016) who observed that wider and shallower residual networks perform better than their deeper and narrower counterpart. On CIFAR-10 and $\mathrm { S V H N ^ { * } }$ , the real-valued representation performs slightly better than its complex counterpart. On CIFAR100, the complex representation outperforms the real one. In general, the obtained results for both representation are quite comparable. To understand the effect of using either real or complex representation for a given task, we designed hybrid models that combine both. Table 2 contains the results for hybrid models. We can observe in the Table 2 that in cases where complex representation outperformed the real one (wide and shallow on CIFAR-100), combining a real-valued convolutional filter with a complex batch normalization improves the accuracy of the real-valued convolutional model. However, the complex-valued one is still outperforming it. In cases, where real-valued representation outperformed the complex one (wide and shallow on CIFAR-10 and ${ \mathrm { S V H N } } ^ { * }$ ), replacing a complex batch normalization by a regular one increased the accuracy of the complex convolutional model. Despite that replacement, the real-valued model performs better in terms of accuracy for such tasks. In general, these experiments show that the difference in efficiency between the real and complex models varies according to the dataset, to the task and to the architecture.
+
+Ablation studies were performed in order to investigate the importance of the 2D whitening operation that occurs in the complex batch normalization. We replaced the complex batch normalization layers with a naive variant (NCBN) which, instead of left multiplying the centred unit by the inverse square root of its covariance matrix, just divides it by its complex variance. Here, this naive variant of CBN is Mimicking the regular BN by not taking into account correlation between the elements in the complex unit. The Naive variant of the Complex Batch Normalization performed very poorly; In 5 out of 6 experiments, training failed with the appearance of NaNs (See Section 6.6 for the explanation). By way of contrast, all 6 complex-valued Batch Normalization experiments converged. Results are given in Table 2.
+
+Another ablation study was undertaken to compare CReLU, modReLU and $z$ RELU. Again the differences were stark: All CReLU experiments converged and outperformed both modReLU and $z$ ReLU, both which variously failed to converge or fared substantially worse. We think that modRelu didn’t perform as well as CReLU due to the fact that consecutive layers in a feed-forward net do not represent time-sequential patterns, and so, they might need to drop some phase information. Results are reported in Table 6.4. More discussion about phase information encoding is presented in section 6.7.
+
+Table 3: MusicNet experiments. $F S$ is the sampling rate. Params is the total number of parameters. We report the average precision (AP) metric that is the area under the precision-recall curve.   
+
+<table><tr><td>ARCHITECTURE</td><td>FS</td><td>PARAMS</td><td>AP,%</td></tr><tr><td>SHALLOW, REAL</td><td>11kHz</td><td></td><td>66.1</td></tr><tr><td>SHALLOW, COMPLEX</td><td>11kHz</td><td></td><td>66.0</td></tr><tr><td>SHALLOW, THICKSTUN ET AL. (2016)</td><td>44.1kHz</td><td>=</td><td>67.8</td></tr><tr><td>DEEP, REAL</td><td>11kHz</td><td>10.0M</td><td>69.6</td></tr><tr><td>DEEP, COMPLEX</td><td>11kHz</td><td>8.8M</td><td>72.9</td></tr></table>
+
+# 4.2 AUTOMATIC MUSIC TRANSCRIPTION
+
+In this section we present results for the automatic music transcription (AMT) task. The nature of an audio signal allows one to exploit complex operations as presented earlier in the paper. The experiments were performed on the MusicNet dataset (Thickstun et al., 2016). For computational efficiency we resampled the original input from $4 4 . 1 \mathrm { k H z }$ to 11kHz using the algorithm described in Smith (2002). This sampling rate is sufficient to recognize frequencies presented in the dataset while reducing computational cost dramatically. We modeled each of the 84 notes that are present in the dataset with independent sigmoids (due to the fact that notes can fire simultaneously). We initialized the bias of the last layer to the value of -5 to reflect the distribution of silent/non-silent notes. As in the baseline, we performed experiments on the raw signal and the frequency spectrum. For complex experiments with the raw signal, we considered its imaginary part equal to zero. When using the spectrum input we used its complex representation (instead of only the magnitudes, as usual for AMT) for both real and complex models. For the real model, we considered the real and imaginary components of the spectrum as separate channels. The model we used for raw signals is a shallow convolutional network similar to the model used in the baseline, with the size reduced by a factor of 4 (corresponding to the reduction of the sampling rate). The filter size was 512 samples (about $1 2 \mathrm { m s } \mathrm { \Omega }$ ) with a stride of 16. The model for the spectral input is similar to the VGG model (Simonyan and Zisserman, 2015). The first layer has filter with size of 7 and is followed by 5 convolutional layers with filters of size 3. The final convolution block is followed by a fully connected layer with 2048 units. The latter is followed, in its turn, by another fully connected layer with 84 sigmoidal units. In all of our experiments we use an input window of 4096 samples or its corresponding FFT (which corresponds to the 16,384 window used in the baseline) and predicted notes in the center of the window. All networks were optimized with Adam. We start our learning rate at $1 0 ^ { - 3 }$ for the first 10 epochs and then anneal it by a factor of 10 at each of the epochs 100, 120 and 150. We end the training at epoch 200. For the real-valued models, we have used ReLU as activation. CReLU has been used as activation for the complex-valued models.
+
+The complex network was initialized using the unitary initialization scheme respecting the He criterion as described in Section 3.6. For the real-valued network, we have used the analogue initial√ization of the weight tensor. It consists of performing an orthogonal initialization with a gain of $\sqrt { 2 }$ . The complex batch normalization was applied according to Section 3.5. Following Thickstun et al. (2016) we used recordings with ids $, 2 3 0 3 ^ { \prime }$ , ’2382’, ’1819’ as the test subset and additionally we created a validation subset using recording ids ’2131’, ’2384’, ’1792’, ’2514’, ’2567’, ’1876’ (randomly chosen from the training set). The validation subset was used for model selection and early stopping. The remaining 321 files were used for training. The results are summarized on Table 3. We achieve a performance comparable to the baseline with the shallow convolutional network. our VGG-based deep real-valued model reaches $6 9 . 6 \%$ average precision on the downsampled data. With significantly fewer parameters than its real counterpart, the VGG-based deep complex model, achieves $7 2 . 9 \%$ average precision which is the state of the art to the best of our knowledge. See Figures 2 and 3 in the Appendix for precision-recall curves and a sample of the output of the model.
+
+Table 4: Speech Spectrum Prediction on TIMIT test set. CConv-LSTM denotes the Complex Convolutional LSTM.   
+
+<table><tr><td>MODEL</td><td>#PARAMS</td><td>MSE(VALIDATION)</td><td>MSE(TEST)</td></tr><tr><td>LSTM WISDOM ET AL. (2016)</td><td>~135K</td><td>16.59</td><td>16.98</td></tr><tr><td>FULL-CAPACITY URNN WISDOM ET AL.(2016)</td><td>~135K</td><td>14.56</td><td>14.66</td></tr><tr><td>CONV-LSTM (OUR BASELINE)</td><td>≈88K</td><td>11.10</td><td>12.18</td></tr><tr><td>CCONV-LSTM (OURS)</td><td>~88K</td><td>10.78</td><td>11.90</td></tr></table>
+
+# 4.3 SPEECH SPECTRUM PREDICTION
+
+We apply both a real Convolutional LSTM Xingjian et al. (2015) and a complex Convolutional LSTM on speech spectrum prediction task (See section 6.5 in the Appendix for the details of the real and complex Convolutional LSTMs). In this task, the model predicts the magnitude spectrum. It implicitly infers the real and imaginary components of the spectrum at time $t + 1$ , given all the spectrum (imaginary part and real components) up to time $t$ . This is slightly different from (Wisdom et al., 2016). The real and imaginary components are considered as separate channels in both model. We evaluate the model with mean-square-error (MSE) on log-magnitude to compare with the others Wisdom et al. (2016). The experiments are conducted on a downsampled (8kHz) version of the TIMIT dataset. By following the steps in Wisdom et al. (2016), raw audio waves are transformed into frequency domain via short-time Fourier transform (STFT) with a Hann analysis window of 256 samples and a window hop of 128 samples $5 0 \%$ overlap). We use a training set with 3690 utterances, a validation set with 400 utterances and a standard test set with 192 utterance.
+
+To match the number of parameters for both model, the Convolutional LSTM has 84 feature maps while the complex model has 60 complex feature maps (120 feature maps in total). Adam Kingma and Ba (2014) with a fixed learning rate of 1e-4 is used in both experiments. We initialize the complex model with the unitary initialization scheme and the real model with orthogonal initialization respecting the Glorot criterion. The result is shown in Table 4 and the learning curve is shown in Figure 4. Our baseline model has achieved the state of the art and the complex convolutional LSTM model performs better over the baseline in terms of MSE and convergence.
+
+# 5 CONCLUSIONS
+
+We have presented key building blocks required to train complex valued neural networks, such as complex batch normalization and complex weight initialization. We have also explored a wide variety of complex convolutional network architectures, including some yielding competitive results for image classification and state of the art results for a music transcription task and speech spectrum prediction. We hope that our work will stimulate further investigation of complex valued networks for deep learning models and their application to more challenging tasks such as generative models for audio and images.
+
+# ACKNOWLEDGEMENTS
+
+We are grateful to Roderick Murray-Smith, Jörn-Henrik Jacobsen, Jesse Engel and all the students at MILA, especially Jason Jo, Anna Huang and Akram Erraqabi for helpful feedback and discussions. We also thank the developers of Theano (Theano Development Team, 2016) and Keras (Chollet et al., 2015). We are grateful to Samsung and the Fonds de Recherche du Québec – Nature et Technologie for their financial support. We would also like to acknowledge NVIDIA for donating a DGX-1 computer used in this work.
+
+# REFERENCES
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015.
+
+Tim Salimans and Diederik P Kingma. Weight normalization: A simple reparameterization to accelerate training of deep neural networks. arXiv preprint arXiv:1602.07868, 2016.
+
+Rupesh K Srivastava, Klaus Greff, and Jürgen Schmidhuber. Training very deep networks. In Advances in neural information processing systems, pages 2377–2385, 2015.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. arXiv preprint arXiv:1512.03385, 2015a.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. arXiv preprint arXiv:1603.05027, 2016.
+
+Sepp Hochreiter. Untersuchungen zu dynamischen neuronalen Netzen. PhD thesis, diploma thesis, institut für informatik, lehrstuhl prof. brauer, technische universität münchen, 1991.
+
+T Nitta. On the critical points of the complex-valued neural network. In Neural Information Processing, 2002. ICONIP’02. Proceedings of the 9th International Conference on, volume 3, pages 1099–1103. IEEE, 2002.
+
+Akira Hirose and Shotaro Yoshida. Generalization characteristics of complex-valued feedforward neural networks in relation to signal coherence. IEEE Transactions on Neural Networks and learning systems, 23(4): 541–551, 2012.
+
+Martin Arjovsky, Amar Shah, and Yoshua Bengio. Unitary evolution recurrent neural networks. arXiv preprint arXiv:1511.06464, 2015.
+
+Ivo Danihelka, Greg Wayne, Benigno Uria, Nal Kalchbrenner, and Alex Graves. Associative long short-term memory. arXiv preprint arXiv:1602.03032, 2016.
+
+Scott Wisdom, Thomas Powers, John Hershey, Jonathan Le Roux, and Les Atlas. Full-capacity unitary recurrent neural networks. In Advances in Neural Information Processing Systems, pages 4880–4888, 2016.
+
+Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8):1735–1780, 1997.
+
+Tony A Plate. Holographic reduced representation: Distributed representation for cognitive structures. 2003.
+
+David P Reichert and Thomas Serre. Neuronal synchrony in complex-valued deep networks. arXiv preprint arXiv:1312.6115, 2013.
+
+Aaron van den Oord, Nal Kalchbrenner, Lasse Espeholt, Oriol Vinyals, Alex Graves, et al. Conditional image generation with pixelcnn decoders. In Advances In Neural Information Processing Systems, pages 4790– 4798, 2016a.
+
+Aäron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. CoRR abs/1609.03499, 2016b.
+
+Kyunghyun Cho, Bart Van Merriënboer, Dzmitry Bahdanau, and Yoshua Bengio. On the properties of neural machine translation: Encoder-decoder approaches. arXiv preprint arXiv:1409.1259, 2014.
+
+Julian Georg Zilly, Rupesh Kumar Srivastava, Jan Koutník, and Jürgen Schmidhuber. Recurrent highway networks. arXiv preprint arXiv:1607.03474, 2016.
+
+Guangji Shi, Maryam Modir Shanechi, and Parham Aarabi. On the importance of phase in human speech recognition. IEEE transactions on audio, speech, and language processing, 14(5):1867–1874, 2006.
+
+Alan V Oppenheim and Jae S Lim. The importance of phase in signals. Proceedings of the IEEE, 69(5): 529–541, 1981.
+
+Oren Rippel, Jasper Snoek, and Ryan P Adams. Spectral representations for convolutional neural networks. In Advances in Neural Information Processing Systems, pages 2449–2457, 2015.
+
+George M Georgiou and Cris Koutsougeras. Complex domain backpropagation. IEEE transactions on Circuits and systems II: analog and digital signal processing, 39(5):330–334, 1992.
+
+Richard S Zemel, Christopher KI Williams, and Michael C Mozer. Lending direction to neural networks. Neural Networks, 8(4):503–512, 1995.
+
+Taehwan Kim and Tülay Adalı. Approximation by fully complex multilayer perceptrons. Neural computation, 15(7):1641–1666, 2003.
+
+Akira Hirose. Complex-valued neural networks: theories and applications, volume 5. World Scientific, 2003.
+
+Tohru Nitta. Orthogonality of decision boundaries in complex-valued neural networks. Neural Computation, 16(1):73–97, 2004.
+
+Geoffrey E Hinton, Simon Osindero, and Yee-Whye Teh. A fast learning algorithm for deep belief nets. Neural computation, 18(7):1527–1554, 2006.
+
+Yoshua Bengio, Pascal Lamblin, Dan Popovici, Hugo Larochelle, et al. Greedy layer-wise training of deep networks. Advances in neural information processing systems, 19:153, 2007.
+
+Christopher Poultney, Sumit Chopra, Yann L Cun, et al. Efficient learning of sparse representations with an energy-based model. In Advances in neural information processing systems, pages 1137–1144, 2007.
+
+Joan Bruna, Soumith Chintala, Yann LeCun, Serkan Piantino, Arthur Szlam, and Mark Tygert. A mathematical motivation for complex-valued convolutional networks. arXiv preprint arXiv:1503.03438, 2015.
+
+Edouard Oyallon and Stéphane Mallat. Deep roto-translation scattering for object classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2865–2873, 2015.
+
+Mark Tygert, Arthur Szlam, Soumith Chintala, Marc’Aurelio Ranzato, Yuandong Tian, and Wojciech Zaremba. Scale-invariant learning and convolutional networks. arXiv preprint arXiv:1506.08230, 2015.
+
+Daniel E Worrall, Stephan J Garbin, Daniyar Turmukhambetov, and Gabriel J Brostow. Harmonic networks: Deep translation and rotation equivariance. arXiv preprint arXiv:1612.04642, 2016.
+
+Théo Trouillon, Johannes Welbl, Sebastian Riedel, Éric Gaussier, and Guillaume Bouchard. Complex embeddings for simple link prediction. In International Conference on Machine Learning, pages 2071–2080, 2016.
+
+Théo Trouillon and Maximilian Nickel. Complex and holographic embeddings of knowledge graphs: a comparison. arXiv preprint arXiv:1707.01475, 2017.
+
+Andy M Sarroff, Victor Shepardson, and Michael A Casey. Learning representations using complex-valued nets. arXiv preprint arXiv:1511.06351, 2015.
+
+Nitzan Guberman. On complex valued convolutional neural networks. arXiv preprint arXiv:1602.09046, 2016.
+
+Michael Cogswell, Faruk Ahmed, Ross Girshick, Larry Zitnick, and Dhruv Batra. Reducing overfitting in deep networks by decorrelating representations. arXiv preprint arXiv:1511.06068, 2015.
+
+Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1): 1929–1958, 2014.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Aistats, volume 9, pages 249–256, 2010.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing humanlevel performance on imagenet classification. In Proceedings of the IEEE international conference on computer vision, pages 1026–1034, 2015b.
+
+Jonathan Tompson, Ross Goroshin, Arjun Jain, Yann LeCun, and Christoph Bregler. Efficient object localization using convolutional networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 648–656, 2015.
+
+Yurii Nesterov. A method of solving a convex programming problem with convergence rate o (1/k2). 1983.
+
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016.
+
+John Thickstun, Zaid Harchaoui, and Sham Kakade. Learning features of music from scratch. In Proc. ICLR, 2016.
+
+Julius O Smith. Digital audio resampling. Online http://www-ccrma. stanford. edu/˜ jos/resample, 2002.
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In Proc. ICLR, 2015.
+
+SHI Xingjian, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional lstm network: A machine learning approach for precipitation nowcasting. In Advances in neural information processing systems, pages 802–810, 2015.   
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   
+Theano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv e-prints, abs/1605.02688, May 2016. URL http://arxiv.org/abs/1605.02688.   
+François Chollet et al. Keras: Deep learning library for theano and tensorflow. URL: https://keras. io/k, 2015.
+
+# 6 APPENDIX
+
+In practice, the complex convolution operation is implemented as illustrated in Fig.1a where $M _ { I }$ , $M _ { R }$ refer to imaginary and real feature maps and $K _ { I }$ and $K _ { R }$ refer to imaginary and real kernels. ${ M } _ { I } K _ { I }$ refers to result of a real-valued convolution between the imaginary kernels $K _ { I }$ and the imaginary feature maps $M _ { I }$ .
+
+![](images/92dccb0f0918ebde426f009043a7902b44be72ec1542c07d1663e151cc742266.jpg)  
+(b) A complex convolutional residual network (left) and an equivalent real-valued residual network (right).
+
+![](images/6b0385d4f5f688cf1527654f2b7e3c330c71e8c9db847cc38c552264c78306d2.jpg)  
+Figure 1: Complex convolution and residual network implementation details.
+
+(a) An illustration of the complex convolution operator.
+
+# 6.1 MUSICNET ILLUSTRATIONS
+
+![](images/c30f563756156611c63f4b25c9c460ba693709c9ac386f319cddfef8902fbcca.jpg)  
+Figure 2: Precision-recall curve
+
+![](images/3224962eda8a53257b443ae4f53948531ca7533953b9748ab02a2efea1c1f19e.jpg)  
+Figure 3: Predictions (Top) vs. ground truth (Bottom) for a music segment from the test set.
+
+# 6.2 HOLOMORPHISM AND CAUCHY–RIEMANN EQUATIONS
+
+Holomorphism, also called analyticity, ensures that a complex-valued function is complex differentiable in the neighborhood of every point in its domain. This means that the derivative, $f ^ { \prime } ( z _ { 0 } ) \equiv$ $\begin{array} { r } { \operatorname* { l i m } _ { \Delta z  0 } \big [ \frac { ( f ( z _ { 0 } ) + \bar { \Delta } z ) - f ( z _ { 0 } ) } { \Delta z } \big ] } \end{array}$ of le $f$ very poinsuch that $z _ { \mathrm { 0 } }$ $f$ er. $f$ is and complex-valuedare real-valued $z = x + i y$ $f ( z ) = u ( x , y ) + i v ( x , y )$ $u$ $v$ functions. One possible way of expressing $\Delta z$ is to have $\Delta z = \Delta x + i \Delta y$ . $\Delta z$ can approach 0 from multiple directions (along the real axis, imaginary axis or in-between). However, in order to be complex differentiable, $f ^ { \prime } ( z _ { 0 } )$ must be the same complex quantity regardless of direction of approach. When $\Delta z$ approaches 0 along the real axis, $f ^ { \prime } ( z _ { 0 } )$ could be written as:
+
+$$
+\begin{array} { r } { f ^ { \prime } ( z _ { 0 } ) \equiv \underset { \Delta z  0 } { \mathrm { l i m } } [ \frac { ( f ( z _ { 0 } ) + \Delta z ) - f ( z _ { 0 } ) } { \Delta z } ] } \\ { = \underset { \Delta x  0 } { \mathrm { l i m } } \underset { \Delta y  0 } { \mathrm { l i m } } [ \frac { \Delta u ( x _ { 0 } , y _ { 0 } ) + i \Delta v ( x _ { 0 } , y _ { 0 } ) } { \Delta x + i \Delta y } ] } \\ { = \underset { \Delta x  0 } { \mathrm { l i m } } [ \frac { \Delta u ( x _ { 0 } , y _ { 0 } ) + i \Delta v ( x _ { 0 } , y _ { 0 } ) } { \Delta x + i 0 } ] . } \end{array}
+$$
+
+When $\Delta z$ approaches 0 along the imaginary axis, $f ^ { \prime } ( z _ { 0 } )$ could be written as:
+
+$$
+\begin{array} { r l } & { = \underset { \Delta y  0 } { \operatorname* { l i m } } \underset { \Delta x  0 } { \operatorname* { l i m } } [ \frac { \Delta u ( x _ { 0 } , y _ { 0 } ) + i \Delta v ( x _ { 0 } , y _ { 0 } ) } { \Delta x + i \Delta y } ] } \\ & { = \underset { \Delta y  0 } { \operatorname* { l i m } } [ \frac { \Delta u ( x _ { 0 } , y _ { 0 } ) + i \Delta v ( x _ { 0 } , y _ { 0 } ) } { 0 + i \Delta y } ] } \end{array}
+$$
+
+Satisfying equations 11 and 12 is equivalent of having $\begin{array} { r } { \frac { \partial f } { \partial z } = \frac { \partial u } { \partial x } + i \frac { \partial v } { \partial x } = - i \frac { \partial u } { \partial y } + \frac { \partial v } { \partial y } } \end{array}$ . So, in order to be complex differentiable, $f$ should satisfy $\begin{array} { r } { { \frac { \partial u } { \partial x } } = { \frac { \partial v } { \partial y } } } \end{array}$ and $\begin{array} { r } { \frac { \partial u } { \partial y } = - \frac { \partial v } { \partial x } } \end{array}$ . These are called the Cauchy–Riemann equations and they give a necessary condition for $f$ to be complex differentiable or "holomorphic". Given that $u$ and $v$ have continuous first partial derivatives, the Cauchy-Riemann equations become a sufficient condition for $f$ to be holomorphic.
+
+# 6.3 THE GENRALIZED COMPLEX CHAIN RULE FOR A REAL-VALUED LOSS FUNCTION
+
+If $L$ is a real-valued loss function and $z$ is a complex variable such that $z = x + i y$ where $x , y \in \mathbb { R }$ , then:
+
+$$
+\nabla _ { L } ( z ) = \frac { \partial L } { \partial z } = \frac { \partial L } { \partial x } + i \frac { \partial L } { \partial y } = \frac { \partial L } { \partial \Re ( z ) } + i \frac { \partial L } { \partial \Im ( z ) } = \Re ( \nabla _ { L } ( z ) ) + i \Im \left( \nabla _ { L } ( z ) \right) .
+$$
+
+Now if we have another complex variable $t = r + i s$ where $z$ could be expressed in terms of $t$ and $r , s \in \mathbb { R }$ , we would then have:
+
+$$
+\begin{array} { r l } & { \nabla _ { L } ( t ) = \displaystyle \frac { \partial L } { \partial t } = \frac { \partial L } { \partial r } + i \frac { \partial L } { \partial s } } \\ & { \qquad = \displaystyle \frac { \partial L } { \partial x } \frac { \partial x } { \partial r } + \frac { \partial L } { \partial y } \frac { \partial y } { \partial r } + i \bigg ( \frac { \partial L } { \partial x } \frac { \partial x } { \partial s } + \frac { \partial L } { \partial y } \frac { \partial y } { \partial s } \bigg ) } \\ & { \qquad = \displaystyle \frac { \partial L } { \partial x } \bigg ( \frac { \partial x } { \partial r } + i \frac { \partial x } { \partial s } \bigg ) + \frac { \partial L } { \partial y } \bigg ( \frac { \partial y } { \partial r } + i \frac { \partial y } { \partial s } \bigg ) } \\ & { \qquad = \displaystyle \frac { \partial L } { \partial \Re ( z ) } \bigg ( \frac { \partial x } { \partial r } + i \frac { \partial x } { \partial s } \bigg ) + \frac { \partial L } { \partial \Re ( z ) } \bigg ( \frac { \partial y } { \partial r } + i \frac { \partial y } { \partial s } \bigg ) } \\ & { \qquad = \Re ( \nabla _ { L } ( z ) ) \bigg ( \frac { \partial x } { \partial r } + i \frac { \partial x } { \partial s } \bigg ) + \Im ( \nabla _ { L } ( z ) ) \bigg ( \frac { \partial y } { \partial r } + i \frac { \partial y } { \partial s } \bigg ) . } \end{array}
+$$
+
+# 6.4 COMPUTATIONAL COMPLEXITY AND FLOPS
+
+In terms of computational complexity, the convolutional operation and the complex batchnorm are of the same order as their real counterparts. However, as a complex multiplication is 4 times more expensive than its real counterpart, all complex convolutions are 4 times more expensive as well.
+
+Additionally, the complex BatchNorm is not implemented in cuDNN and therefore had to be simulated with a sizeable sequence of elementwise operations. This leads to a ballooning of the number of nodes in the compute graph and to inefficiencies due to lack of effective operation fusion. A dedicated cuDNN kernel will, however, reduce the cost to little more than that of the real-valued BatchNorm.
+
+Ignoring elementwise operations, which constitute a negligible fraction of the floating-point operations in the neural network, we find that for all architectures in and for all of CIFAR10, CIFAR100 or SVHN, the inference cost in real FLOPS per example is roughly identical. It is $\sim 2 6 5$ MFLOPS for the $\mathbb { R }$ -valued variant and $\sim 1 0 3 0$ MFLOPS for the $\mathbb { C }$ -valued variant of the architecture, approximately quadruple.
+
+# 6.5 CONVOLUTIONAL LSTM
+
+A Convolutional LSTM is similar to a fully connected LSTM. The only difference is that, instead of using matrix multiplications to perform computation, we use convolutional operations. The computation in a realvalued Convolutional LSTM is defined as follows:
+
+$$
+\begin{array} { r l } & { \mathbf i _ { t } = \sigma \big ( \mathbf W _ { x i } * \mathbf x _ { t } + \mathbf W _ { h i } * \mathbf W _ { t - 1 } + \mathbf b _ { i } \big ) } \\ & { \mathbf f _ { t } = \sigma \big ( \mathbf W _ { x f } * \mathbf x _ { t } + \mathbf W _ { h f } * \mathbf h _ { t - 1 } + \mathbf b _ { f } \big ) } \\ & { \mathbf c _ { t } = \mathbf f _ { t } \circ \mathbf c _ { t - 1 } + \mathbf i _ { t } \circ \operatorname { t a n h } ( \mathbf W _ { x c } * \mathbf x _ { t } + \mathbf W _ { h c } * \mathbf h _ { t - 1 } + \mathbf b _ { c } ) } \\ & { \mathbf o _ { t } = \sigma \big ( \mathbf W _ { x o } * \mathbf x _ { t } + \mathbf W _ { h o } * \mathbf h _ { t - 1 } + \mathbf b _ { o } \big ) } \\ & { \mathbf h _ { t } = \mathbf o _ { t } \circ \operatorname { t a n h } ( \mathbf c _ { t } ) } \end{array}
+$$
+
+Where $\sigma$ denotes the sigmoidal activation function, $^ { \circ }$ the elementwise multiplication and $^ *$ the real-valued convolution. $\mathbf { i } _ { t } , \mathbf { f } _ { t }$ , $\mathbf { o } _ { t }$ represent the vector notation of the input, forget and output gates respectively. $\mathbf { c } _ { t }$ and $\mathbf { h } _ { t }$ represent the vector notation of the cell and hidden states respectively. the gates and states in a ConvLSTM are tensors whose last two dimensions are spatial dimensions. For each of the gates, $\mathbf { W } _ { x g a t e }$ and $\mathbf { W } _ { h g a t e }$ are respectively the input and hidden kernels.
+
+For the Complex Convolutional LSTM, we just replace the real-valued convolutional operation by its complex counterpart. We maintain the real-valued elementwise multiplication. The sigmoid and tanh are both performed separately on the real and the imaginary parts.
+
+![](images/09f98ae66ce33ce101927c7cfe89bfa053acc94619999558b8cd236e5d6ba76e.jpg)  
+Figure 4: Learning curve for speech spectrum prediction from dev set.
+
+6.6 COMPLEX STANDARDIZATION AND INTERNAL COVARIATE SHIFT
+
+![](images/71a04361f1805798bedb85de2806000851f4c08e536e0431419569bb3125d5af.jpg)  
+Figure 5: Depiction of Complex Standardization in Deep Complex Networks. At left, Naive Complex Standardization (division by complex standard deviation); At right, Complex Standardization (left-multiplication by inverse square root of covariance matrix between $\Re$ and $\mathfrak { I }$ ). The 250 input complex scalars are at the bottom, with $\Re ( v )$ plotted on $x$ (red axis) and $\Im ( v )$ plotted on $y$ (green axis). Deeper representations correspond to greater $z$ (blue axis). The gray ellipse encloses the input scalars within 1 standard deviation of the mean. Red ellipses enclose all scalars within 1 standard deviation of the mean after “standardization”. Blue ellipses enclose all scalars within 1 standard deviation of the mean after left-multiplying all the scalars by a random $2 \times 2$ linear transformation matrix. With the naive standardization, the distribution becomes progressively more elliptical with every layer, eventually collapsing to a line. This ill-conditioning manifests itself as NaNs in the forward pass or backward pass. With the complex standardization, the points’ distribution is always successfully re-circularized.
+
+# 6.7 PHASE INFORMATION ENCODING
+
+![](images/3b55a6b56be6f20fbe3ba3770fa4db0fa285f12d9420d1f26a402c94bf458299.jpg)
+
+Figure 6: Phase information encoding for each of the activation functions tested for the Deep Complex Network. The $\mathbf { X }$ -axis represents the real part and the y-axis axis represents the imaginary part; The bottom figure corresponds to the case where $b < 0$ for modReLU. The radius of the white circle is equal to $| b |$ . In case where $b \geq 0$ , the whole complex plane would be preserving both phase and magnitude information and the whole plane would have been colored with orange. Different colors represents different encoding of the complex information in the plane. We can see the for both zReLU and modReLU, the complex representation is discriminated into two regions, i.e, the one that preserves the whole complex information (colored in orange) and the one that cancels it (colored in white). However, CReLU discriminates the complex information into 4 regions where in two of which, phase information is projected and not canceled. This allows CReLU to discriminate information easier with respect to phase information than the other activation functions. For both zReLU and modReLU, we can see that phase information may be preserved explicitly through a number of layers when these activation functions are operating in their linear regime, prior to a layer further up in a network where the phase of an input lies in a zero region. CReLU has more flexibility manipulating phase as it can either set it to zero or $\pi / 2$ , or even delete the phase information (when both real and imaginary parts are canceled) at a given level of depth in the network.

parse/train/HkehD3VtvS/HkehD3VtvS.md

@@ -0,0 +1,321 @@
+# DEEP REASONING NETWORKS FOR UNSUPERVISED PATTERN DE-MIXING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We introduce Deep Reasoning Networks (DRNets), an end-to-end framework that combines deep learning with reasoning for solving pattern de-mixing problems, typically in an unsupervised or weakly-supervised setting. DRNets exploit problem structure and prior knowledge by tightly combining logic and constraint reasoning with stochastic-gradient-based neural network optimization. We illustrate the power of DRNets on de-mixing overlapping hand-written Sudokus (MultiMNIST-Sudoku) and on a substantially more complex task in scientific discovery that concerns inferring crystal structures of materials from X-ray diffraction data (Crystal-Structure-Phase-Mapping). DRNets significantly outperform the state of the art and experts’ capabilities on Crystal-Structure-Phase-Mapping, recovering more precise and physically meaningful crystal structures. On Multi-MNISTSudoku, DRNets perfectly recovered the mixed Sudokus’ digits, with $100 \%$ digit accuracy, outperforming the supervised state-of-the-art MNIST de-mixing models.
+
+# 1 INTRODUCTION
+
+Deep learning has achieved tremendous success in areas such as vision, speech recognition, language translation, and autonomous driving. Nevertheless, certain limitations of deep learning are generally recognized, in particular, limitations due to the fact that deep learning approaches heavily depend on the availability of large amounts of labeled data. In certain domains, such as scientific discovery, it is often the case that scientists don’t have large amounts of labeled data and instead have to rely on prior knowledge to make sense of the data. One grand challenge in scientific discovery is to perform high-throughput unsupervised interpretation of scientific data, given its exponential growth in generation rates, dramatically outpacing humans’ ability to analyze them. Herein we consider pattern de-mixing problems, which involve decomposing a mixed signal into the collection of source patterns, such as separating mixtures of X-ray diffraction (XRD) signals into the source XRD signals of the corresponding crystal structures, a key challenge in materials discovery. More generally, pattern de-mixing problems are pervasive in scientific areas as diverse as biology, astronomy, and materials science, as well as in commercial applications for e.g., healthcare and music.
+
+We propose Deep Reasoning Networks (DRNets), an end-to-end framework that combines deep learning with logical and constraint reasoning for solving unsupervised or very-weakly-supervised pattern de-mixing tasks. We illustrate the power of DRNets for disentangling two overlapping handwritten Sudokus (Multi-MNIST-Sudoku) (see Fig.1) and for solving a substantially more complex de-mixing task in scientific discovery that concerns inferring crystal structures of materials from X-ray diffraction data, which we refer to as Crystal-Structure-Phase-Mapping. Both de-mixing tasks require probabilistic reasoning to interpret noisy and uncertain data, while satisfying a set of rules: Sudoku rules and thermodynamic rules, respectively. For example, de-mixing hand written digits is challenging, but it becomes more feasible when we reason about the prior knowledge concerning the two overlapping Sudokus. Crystal structure phase mapping is yet substantially more complex. In fact, crystal structure phase mapping easily becomes too complex for experts to solve and is a major bottleneck in high-throughput materials discovery. DRNets are inspired and motivated by problems from scientific discovery, such as crystal structure phase mapping.
+
+Our contributions: (1) We introduce Deep Reasoning Networks (DRNets), an end-to-end framework that combines deep learning with logical and constraint reasoning for unsupervised or veryweakly-supervised de-mixing tasks. Specifically, DRNets perform end-to-end deep reasoning by encoding a latent space of the input data that captures the structure and prior knowledge constraints within and among data points (Fig.2). The latent space is used by a generative decoder to generate the targeted output, which should be consistent with the input data and prior knowledge. Subsequently, DRNets optimize an objective function capturing the overall problem objective as well as prior knowledge in the form of weighted constraints. (2) To instantiate the logical constraints in DRNets, we introduce a group of entropy-based continuous relaxations that use probabilistic modeling to encode general discrete constraints including sparsity, cardinality and so-called AllDifferent constraints.To optimize those constraints, we introduce a variant of standard SGD method (Robbins & Monro, 1985) called constraint-aware stochastic gradient descent, which batches data points involved in the same constraint component together and dynamically adjust the constraints’ weights as a function of their satisfiability. In the following sections, we show how to encode Multi-MNIST-Sudoku and Crystal-Structure-Phase-Mapping as DRNets, by properly defining the structure of the latent space, additional reasoning modules to model the problem constraints (prior knowledge), and the components of the objective function. De facto, these examples illustrate how to develop “gadgets” to encode a variety of constraints and prior knowledge in DRNets. (3) We demonstrate the potential of DRNets on two de-mixing tasks with detailed experimental results. We show how (3.1) DRNets significantly outperformed the state of the art and human experts on Crystal-Structure-Phase-Mapping instances, recovering more precise, interpretable, and physically meaningful crystal structure pattern decompositions. In this task, DRNets solve a previously unsolved chemical system, which subsequently led to the discovery of a new material that is important for solar fuels technology. (3.2) On Multi-MNIST-Sudoku instances, without direct supervision, DRNets perfectly recovered the digits in the mixed Sudokus with $100 \%$ digit accuracy, outperforming the supervised state-of-the-art MNIST de-mixing models, including CapsuleNet (Sabour et al., 2017) and ResNet (He et al., 2016).
+
+![](images/fda4227729e674ae134bb664f695286972822b57f9b727826f72c9683576f7b7.jpg)  
+Figure 1: (a) Two $4 \mathbf { x } 4$ Sudokus: The cells in each row, column, and any of the four $2 \mathbf { x } 2$ boxes involving the corner cells have non-repeating digits. (b) Two overlapping Sudokus, with a mixture of two digits in each cell: one from 1 to 4 and the other from 5 to 8. In Multi-MNIST-Sudoku, the digits of two overlapping hand written Sudokus (b) have to be de-mixed (as done by DRNets in (c)). (d) The reconstructed overlapping hand written Sudokus from DRNets.
+
+![](images/2260e4f59d0d984ed32b29b9cb47515c5fba6228ddafeeaab3fd547bee19da90.jpg)  
+Figure 2: Deep Reasoning Networks (DRNets) perform end-to-end deep reasoning by encoding a latent space of the input data that captures prior knowledge constraints and is used by a generative decoder to generate the targeted output. (a) Prior knowledge includes prototypes of digits, which are used to pre-train and build the decoder’s generative module, and Sudoku’s rules, which help DRNet reason about the overlapping digits. (b) Reasoning modules batch data points involved in the same constraints (cells in rows, columns, blocks of a Sudoku) together, enforce that the structure of the latent space satisfies prior knowledge, and dynamically adjust the weights of constraints based on their satisfiability. (c) The overall objective combines responses from the generative decoder (thinking fast) and the reasoning modules (thinking slow).
+
+# 2 RELATED WORK
+
+DRNets have been motivated by scientific tasks such as crystal phase mapping that involve identifying or de-mixing patterns in data that satisfy prior scientific knowledge. In general, for such tasks there are no labeled datasets. So our work focus on unsupervised or weakly supervised learning, using prior knowledge.
+
+Most closely related work: Unsupervised or weakly supervised de-mixing approaches. Pattern de-mixing approaches have been developed under the name of source separation in the signal processing community. The unsupervised methods in this area mostly try to solve the de-mixing, which is in general ill-posed, using different regularizations. Among existing methods, recent work for weakly supervised audio source separation (Zhang et al., 2017) is most related to DRNets since they also employed a generative adversarial network (GAN) in their model. However, their model mainly employs the decoder of GAN to discriminate the reality of separated sources, while DRNets only utilize the generator of GAN as the generative model of possible sources. Moreover, the weakly supervised setting in their paper is actually too strong: they need the true labels of mixed sources, which is almost the goal of our tasks, and therefore it is not applicable to our settings. We now consider the state-of-the-art models for the tasks considered in this paper. For Crystal-structurephase-mapping, due to the lack of labeled datasets, existing models (Ermon et al., 2015; Xue et al., 2017; Bai et al., 2017; 2018; Stanev et al., 2018) are mainly based on non-negative matrix factorization (NMF), which is in general unsupervised. Stanev et al. (2018) proposed the NMF- $\mathbf { \nabla } \cdot \mathbf { k }$ algorithm, which applies a customized clustering process over the results of thousands of runs of pure NMF algorithm (Long et al., 2009) to cluster the common phase patterns. However, NMF- $\mathbf { \nabla } \cdot \mathbf { k }$ does not enforce prior knowledge (namely thermodynamic rules) and therefore the solutions produced are often not completly physically meaningful. To address this limitation several approaches have been developed that use external mixed-integer programming modules to interact with the NMF de-mixing module to enforce prior knowledge (Ermon et al., 2015; Bai et al., 2017; 2018). However, the coordination barrier between the NMF de-mixing module and the reasoning module often results in inferiror performance, where the solution satisfies constraints at the cost of huge reconstruction loss. In contrast to existing models, DRNets seamlessly integrate the pattern de-mixing module and the reasoning module, recovering almost exact ground truth decomposition. In our experiments we thoroughly compare DRNets’ performance against the state of the art (IAFD and NMF-k) for crystal-structure pattern de-mixing. MNIST de-mixing was first studied by Hinton et al. in 2000, where the aim is to identify or de-mix overlapping digits coming from the MNIST datasets (LeCun et al., 1998). More recently, it has been tackled with state-of-the-art neural network models such as CapsuleNet (Sabour et al., 2017) and ResNet (He et al., 2016). Existing works concerning this task are mainly in supervised settings, where we have labels of digits for each overlapping image. However, in this paper, we aim to tackle this task in a weakly supervised setting, where we only have access to the prototypes of single digits and the extra Sudoku rules. Due to the lack of existing models with the same setting, we compared DRNets’s performance against the state-of-the-art supervised models (CapsuleNet and ResNet). By utilizing the supervision from prior knowledge and reasoning, we show that DRNets’ outperformed all supervised models with $100 \%$ digit accuracy.
+
+Enhancing deep learning with symbolic prior knowledge. Exploiting problem structure and reasoning about prior knowledge has been of increasing interest to facilitate deep learning (Garcez et al., 2019). In computer vision, symmetry constraints, bone-length constraints and linear constraints were introduced for human pose estimation (Zhou et al., 2017; 2016) and image segmentation (Pathak et al., 2015) to regularize the output and enhance generalization. In natural language processing, Hu et al. (2016a;b) introduced the posterior regularization (Ganchev et al., 2010) framework into deep learning to incorporate rule-based grammatical knowledge using first order logic. Xu et al. (2017) proposed a semantic loss function to enforce propositional logic constraints on the output of neural networks for semi-supervised multi-class classification tasks. Wang et al. (2019) proposed SATNet, which approximately encodes a MAXSAT solver into a neural network layer called SATNet layer, to explicitly learn the logical structures (e.g., parity function and Sudoku) from the labeled training data.
+
+Previous works in this area primarily focus on supervised or semi-supervised settings for data-rich domains, where direct supervision from labels reduce the importance of explicitly reasoning about prior knowledge. In contrast, with an unsupervised setting, the supervision of DRNets comes from reasoning about prior knowledge and self-reconstruction, which is strongly desired for problems in scientific discovery due to the lack of labeled datasets, and strongly motivated by extensive prior knowledge from sources ranging from fundamental principles to the intuitive experience of scientists.
+
+Among existing works, SATNet is mostly related to DRNets in the sense of bridging logical reasoning with deep learning. However, SATNet is essentially designed for learning logical structures (prior knowledge) from labeled training examples while DRNets aim to facilitate unsupervised learning with known logical constraints. In terms of the encoding of the reasoning module, the semantic loss (Xu et al., 2017) is mostly related to ours. However, the semantic loss encodes constraints by propositional logic, which requires enumerating all possible Boolean assignments that satisfy the constraints. Consequently, the semantic loss has to enumerate a large number of assignments to encode constraints such as $\mathbf { k }$ -sparsity constraints and All-Different constraints, which is not applicable to tasks considered in this paper.
+
+# 3 DEEP REASONING NETWORKS
+
+![](images/e766d310d423e1da7a432a756bfdd4611ad3ec5b5aceaaca63a07a5acd8a4801.jpg)  
+Figure 3: The reduction flow of Deep Reasoning Networks.
+
+DRNets (see Fig.2) are inspired by human thinking (Shivhare & Kumar, 2016): we abstract patterns to higher-level descriptions and combine them with prior-knowledge to fill-in the gaps. Consider the Multi-MNIST-Sudoku example (Fig.1): we first guess the digits in each cell based on the patterns; we re-adjust our initial beliefs and re-image the overlapping patterns by reasoning about Sudoku rules and comparing them to the original ones, potentially involving several iterations. Analogously, in a reasoning system, an inference procedure derives what follows from an initial set of axioms and rules. For example, in a standard $9 \mathrm { x } 9$ Sudoku, an inference procedure identifies the missing cell values of the input Sudoku. A constraint solver is a particular type of reasoning system in which axioms and rules are expressed as constraints and the inference procedure is a search method. Formally, DRNets formulate unsupervised pattern de-mixing as a data-driven constrained optimization, incorporating abstractions and reasoning about structure and prior knowledge:
+
+$$
+\operatorname* { m i n } _ { \theta } \ \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathcal { L } ( G ( \phi _ { \theta } ( \mathbf { x } _ { i } ) ) , \mathbf { x } _ { i } ) \quad \mathrm { ~ s . t . ~ } \phi _ { \theta } ( \mathbf { x } _ { i } ) \in \Omega ^ { \mathrm { l o c a l } } \mathrm { ~ a n d ~ } ( \phi _ { \theta } ( \mathbf { x } _ { 1 } ) , . . . , \phi _ { \theta } ( \mathbf { x } _ { N } ) ) \in \Omega ^ { \mathrm { g l o b a l } }
+$$
+
+In this formulation, $\mathbf { x } _ { i } \in R ^ { n }$ is the $i$ -th $n$ -dimensional input data point, $\phi _ { \theta } ( \cdot )$ is the function of the encoder in DRNets parameterized by $\theta$ , $G ( \cdot )$ denotes the generative decoder, $\dot { \mathcal { L } } ( \cdot , \cdot )$ is the loss function (e.g., evaluating the reconstruction of patterns), $\Omega ^ { \mathrm { l o c a l } }$ and $\Omega ^ { \mathrm { g l o b a l } }$ are the constrained spaces w.r.t. a single input data point and several input data points, respectively. $G ( \cdot )$ is in general a fixed pre-trained or parametric model. For example, in Multi-MNIST-Sudoku, $G ( \cdot )$ is a pre-trained conditional GAN (Mirza & Osindero, 2014) using hand-written digits, and for Crystal-Structure-Phase-Mapping, $G ( \cdot )$ is a Gaussian Mixture model. Note that constraints can involve several (potentially all) data points: e.g., in Sudoku, all digits should form a valid Sudoku and in crystal-structure-phase-mapping, all data points in a composition graph should form a valid phase diagram. Thus, we specify local and global constraints in DRNets – local constraints only involve a single input data point whereas global constraints involve several input data points, and they are optimized using different strategies.
+
+Solving the constrained optimization problem (1) directly is extremely challenging since the objective function in general involves deep neural networks, which are highly non-linear and non-convex, and prior knowledge often even involves combinatorial constraints (Fig.3). Therefore, we use Lagrangian relaxation to approximate equation (1) with an unconstrained optimization problem, i.e.,
+
+$$
+\operatorname* { m i n } _ { \theta } \ \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathcal { L } ( G ( \phi _ { \theta } ( \mathbf { x } _ { i } ) ) , \mathbf { x } _ { i } ) + \lambda ^ { l } \psi ^ { l } ( \phi _ { \theta } ( \mathbf { x } _ { i } ) ) + \sum _ { j = 1 } ^ { N _ { g } } \lambda _ { j } ^ { g } \psi _ { j } ^ { g } ( \{ \phi _ { \theta } ( \mathbf { x } _ { k } ) | k \in S _ { j } \} )
+$$
+
+$N$ is the number of input data points, $N _ { g }$ denotes the number of global constraints, $S _ { j }$ denotes the set of indices w.r.t. the data points involved in the $j$ -th global constraint, and $\psi ^ { l } , \psi _ { j } ^ { g }$ denote the penalty functions for local constraints and global constraints, respectively, along with their corresponding penalty weights $\lambda ^ { l }$ and $\lambda _ { j } ^ { g }$ . In the following, we propose two mechanisms to tackle the above unconstrained optimization task (Fig.3).
+
+Continuous Relaxation: Prior knowledge often involves combinatorial constraints with discrete variables that are difficult to optimize in an end-to-end manner using gradient-based methods. Therefore, we need to design proper continuous relaxations for discrete constraints to make the overall objective function differentiable. Existing works (Hu et al., 2016a; Xu et al., 2017) proposed several relaxations for injecting first-order logic and propositional logic into deep learning. However, limited by the expressive power of those logic formulas, we need a large number of logical terms to express constraints such as $\mathbf { k }$ -sparsity constraints or All-Different constraints. Therefore, to instantiate DRNets for our tasks, we propose a group of entropy-based continuous relaxations to encode general discrete constraints such as sparsity, cardinality and All-Different constraints (see Fig.4). We construct continuous relaxations based on probabilistic modelling of discrete variables, where we model a probability distribution over all possible values for each discrete variable. For example, in Multi-MNIST-Sudoku, a way of encoding the possible two digits in the cell indicated by data point $x _ { i }$ (one from $\{ 1 . . . 4 \}$ and the other from $\{ 5 . . . 8 \}$ ), is to use 8 binary variables $e _ { i , j } \in \{ 0 , 1 \}$ , while requiring $\textstyle \sum _ { j = 1 } ^ { 4 } e _ { i , j } = 1$ and $\textstyle \sum _ { j = 5 } ^ { 8 } e _ { i , j } = 1$ . In DRNets, we model probability distribution $P _ { i }$ and $Q _ { i }$ over digits 1 to 4 and 5 to 8 respectively: $P _ { i , j } , j { = } 1 { \ldots } 4$ and $Q _ { i , j } , j { = } 1 { \ldots } 4$ denote the probability of digit $j$ and the probability of digit $j + 4$ , respectively. We approximate the cardinality constraint of $e _ { i , j }$ by minimizing the entropy of $P _ { i }$ and $Q _ { i }$ , which encourages $P _ { i }$ and $Q _ { i }$ to collapse to one value. Another combinatorial constraint in Multi-MNIST-Sudoku is the All-Different constraint, where all the cells in a constrained set $S$ , i.e., each row, column, and any of four $2 \mathbf { x } 2$ boxes involving the corner cells, must be filled with non-repeating digits. For a probabilistic relaxation of the All-Different constraint, we analogously define the entropy of the averaged digit distribution for all cells in a constrained set $S$ , i.e., $H ( { \bar { P } } _ { S } )$ :
+
+Figure 4: Examples of continuous relaxations: $e _ { i , j } , P _ { i } , Q _ { i } , P _ { M }$ denote binary variables, the discrete distribution over digits 1 to 4, the discrete distribution over digits 5 to 8, and the discrete distribution over values 1 to $M$ .   
+
+<table><tr><td rowspan=1 colspan=1>Cardinality Constraintei,j ∈{0,1} j=1...8s.t.∑=1eij=1and∑j=5eij=1</td><td rowspan=1 colspan=1>Cardinality Constraint Relaxationmin H(Pi)+H(Qi)Θ=-∑=1Pi,jlogPi,j-∑=1Qi,jlogQi,j</td></tr><tr><td rowspan=1 colspan=1>All-Different ConstraintFor all constrained set Ss.t.∑i∈s ei,j = 1 forj = 1...8</td><td rowspan=1 colspan=1>All-Different Constraint RelaxationFor all constrained set SmaxH(Ps)+H(Qs)Θ</td></tr><tr><td rowspan=1 colspan=1>k-Sparsity Constrainteij ∈ {0,1} j=1..M s.t.∑-1eij≤k</td><td rowspan=1 colspan=1>k-SparsityConstraintRelaxationmin max{H(Pm),c}，where c &lt;log k</td></tr></table>
+
+$$
+H ( \hat { P } _ { S } ) = - \sum _ { j = 1 } ^ { 4 } \hat { P } _ { S , j } \log \hat { P } _ { S , j } = - \sum _ { j = 1 } ^ { 4 } \left( \frac { 1 } { | S | } \sum _ { i \in S } P _ { i , j } \right) \log \left( \frac { 1 } { | S | } \sum _ { i \in S } P _ { i , j } \right)
+$$
+
+In this equation, a larger value implies that the digits in the cells of $S$ distribute more uniformly. Thus, we can analogously approximate All-Different constraints by maximizing $H ( \bar { P } _ { S } ) _ { - }$ and $H ( { \bar { Q } } _ { S } )$ . One can see, by minimizing all $H ( P _ { i } )$ and $H ( Q _ { i } )$ to 0 as well as maximizing all $H ( \bar { P } _ { S } )$ and $H ( Q _ { S } )$ to $\log | S |$ , we find a valid solution for the two 4x4 Sudoku puzzles, where all $P _ { i , j }$ are either 0 or 1.
+
+We also relax $k$ -sparsity constraints, which for example in Crystal-Phase-Mapping state the maximum number $k$ of pure phases in an XRD-pattern, by minimizing the entropy of the phase distribution $P _ { M }$ below a threshold $c < \log k$ . We choose the threshold $c < \log k$ because the entropy of a discrete distribution $P _ { M }$ concentrated on at most $k$ values cannot exceed $\log k$ . Note that other relaxations can be adapted in DRNets, for these and other tasks. See also additional relaxations (e.g., for SAT constraints), detailed relaxation derivations, and implementation details in supplementary materials.
+
+Constraint-Aware Stochastic Gradient Descent: We introduce a variant of standard SGD method called constraint-aware SGD, which is conceptually similar to the optimization process in GraphRNN (You et al., 2018), to tackle the optimization of global penalty functions $\psi _ { j } ^ { \dot { g } } ( \{ \phi _ { \underline { { \theta } } } ( \mathbf { x } _ { k } ) | k \ \in \ S _ { j } \} )$ , which involve several (potentially all) data points. We define a constraint graph, an undirected
+
+Input: (i) Data points $\{ x _ { i } \} _ { i = 1 } ^ { N }$ . (ii) Constraint graph. (iii) Penalty functions $\psi ^ { l } ( \cdot )$ and $\psi _ { j } ^ { g } ( \cdot )$ for the local and the global constraints. (iv) Pre-trained or parametric generative decoder $G \bar { ( \cdot ) }$ . 1: Initialize the penalty weights $\lambda ^ { l } , \lambda _ { j } ^ { g }$ and thresholds for all constraints.   
+2: for number of optimization iterations do   
+3: Batch data points $\{ \mathbf { x } _ { 1 } , . . . , \mathbf { x } _ { m } \}$ from the sampled (maximal) connected components. 4: Collect the global penalty functions $\{ \psi _ { j } ^ { g } ( \cdot ) \} _ { j = 1 } ^ { \hat { M } }$ concerning those data points.   
+5: Compute the latent space $\{ \phi _ { \theta } ( \mathbf { x } _ { 1 } ) , . . . , \bar { \phi } _ { \theta } ( \mathbf { x } _ { m } ) \}$ from the encoder.   
+6: Adjust the penalty weights $\lambda _ { l } , \lambda _ { j } ^ { g }$ and thresholds accordingly.   
+7: minimize $\begin{array} { r } { \frac { 1 } { m } \big ( \sum _ { i = 1 } ^ { m } \mathcal { L } ( G ( \phi _ { \theta } ( \mathbf { x } _ { i } ) ) , \mathbf { x } _ { i } ) + \lambda _ { l } \psi ^ { l } ( \phi _ { \theta } ( \mathbf { x } _ { i } ) ) \big ) + \sum _ { j = 1 } ^ { M } \lambda _ { j } ^ { g } \psi _ { j } ^ { g } ( \{ \phi _ { \theta } ( \mathbf { x } _ { k } ) | k \ \in \ S _ { j } \} ) } \end{array}$ using any standard gradient-based optimization method and update the parameters $\theta$ . 8: end for
+
+graph in which each data point forms a vertex and two data points are linked if they are in the same global constraint. Constraint-aware SGD batches data points from the randomly sampled (maximal) connected components in the constraint graph, and optimizes the objective function w.r.t. the subset of global constraints concerning those data points and the associated local constraints. For example, in Multi-MNIST-Sudoku, each overlapping Sudoku forms a maximal connected component, we batch the data points from several randomly sampled overlapping Sudokus and optimize the All-Different constraints (global) as well as the cardinality constraints (local) within them. However, in Crystal-Structure-Phase-Mapping, the maximal connected component becomes too large to batch together, due to the constraints (phase field connectivity and Gibbs-alloying rule) concerning all data points in the composition graph. Thus, we instead only batch a subset (still a connected component) of the maximal connected component – e.g., a path in the composition graph, and optimize the objective function that only concerns constraints within the subset (along the path). By iteratively solving sampled local structures of the ”large” maximal component, we cost-efficiently approximate the entire global constraint. Moreover, for optimizing the overall objective, constraint-aware SGD dynamically adjusts the thresholds and the weights of constraints according to their satisfiability, which can involve non-differentiable functions (See details in appendix). For efficiency and potential capability of generalization, DRNets solve all instances together using constraint-aware SGD (see Algorithm 2).
+
+Algorithm 1 Constraint-aware stochastic gradient descent optimization of deep reasoning networks.   
+
+<table><tr><td rowspan=2 colspan=1>Method</td><td rowspan=1 colspan=2>Accuracy(%)</td><td rowspan=2 colspan=1>Time</td></tr><tr><td rowspan=1 colspan=1>Digit</td><td rowspan=1 colspan=1>Sudoku</td></tr><tr><td rowspan=1 colspan=1>DRNets (Optimization w/Restart)</td><td rowspan=1 colspan=1>100.0</td><td rowspan=1 colspan=1>100.0</td><td rowspan=1 colspan=1>50min</td></tr><tr><td rowspan=1 colspan=1>DRNets (Optimization)</td><td rowspan=1 colspan=1>99.9</td><td rowspan=1 colspan=1>98.6</td><td rowspan=1 colspan=1>28min</td></tr><tr><td rowspan=1 colspan=1>DRNets (Optimization w/oReasoning)</td><td rowspan=1 colspan=1>88.8</td><td rowspan=1 colspan=1>15.0</td><td rowspan=1 colspan=1>110min</td></tr><tr><td rowspan=1 colspan=1>DRNets (Generalization)</td><td rowspan=1 colspan=1>98.0</td><td rowspan=1 colspan=1>75.7</td><td rowspan=1 colspan=1>13min+4hrs</td></tr><tr><td rowspan=1 colspan=1>CapsuleNet</td><td rowspan=1 colspan=1>97.9</td><td rowspan=1 colspan=1>50.9</td><td rowspan=1 colspan=1>1min+30min</td></tr><tr><td rowspan=1 colspan=1>CapsuleNet + local search</td><td rowspan=1 colspan=1>97.9</td><td rowspan=1 colspan=1>57.8</td><td rowspan=1 colspan=1>3hrs+30mins</td></tr><tr><td rowspan=1 colspan=1>ResNet-18</td><td rowspan=1 colspan=1>97.7</td><td rowspan=1 colspan=1>68.5</td><td rowspan=1 colspan=1>3min+10hrs</td></tr><tr><td rowspan=1 colspan=1>ResNet-18 + local search</td><td rowspan=1 colspan=1>97.7</td><td rowspan=1 colspan=1>88.3</td><td rowspan=1 colspan=1>3hrs+10hrs</td></tr></table>
+
+# 4 EXPERIMENTS
+
+We illustrate the power of DRNets mainly on two pattern de-mixing tasks – disentangling two overlapping hand-written Sudokus (Multi-MNIST-Sudoku) and inferring crystal structures of materials from X-ray diffraction data (Crystal-Structure-Phase-Mapping). Limited by the space, we put the details of the experiments and the experimental results of DRNets on other tasks in supplementary material. Note that, since DRNets are an unsupervised framework, we can apply the restart (Gomes et al., 1998) mechanism, i.e., we can re-run DRNets for unsolved instances.
+
+![](images/8007918e5289e28944758067a28340c94422f8981ddd435e6967db73a2ab24d6.jpg)  
+Figure 5: Left: The latent space of DRNets for Multi-MNIST-Sudoku. Right: Accuracy comparison. We show ”test time $^ +$ training time” for supervised baselines and the generalization mode of DRNet, and ”solving time” for the optimization mode of DRNets. (See also supplementary materials.)
+
+Multi-MNIST-Sudoku: We generated 160,000 input data points for each training set, validation set and test set, where each data point corresponds to a $3 2 \mathbf { x } 3 2$ image of overlapping digits coming from MNIST (LeCun et al., 1998) and every 16 data points form a 4-by-4 overlapping Sudokus. For Multi-MNIST-Sudoku, DRNets batch every 16 data points together to enforce the All-Different constraints among the cells of each Sudoku. The encoder of DRNets is composed of two ResNet-18 He et al. (2016) and we use a conditional GAN (Mirza & Osindero, 2014) as our generative decoder (denoted as $G ( \cdot ) )$ ), which is trained using the digits in the training set of MNIST. For each cell $\mathbf { x } _ { i }$ , the encoder encodes a latent space, which consists of two parts: The first part includes two distribution $P _ { i }$ and $Q _ { i }$ (see Fig.5) concerning the possible digits in the cell, and the second part is the latent encodings $z _ { i , 1 } , . . . , z _ { i , 8 }$ of each possible digit conditioned on the overlapping digits, which is used by the generative decoder to generate the corresponding digits $G ( z _ { i , j } )$ . We estimate the two digits in the cell by computing the expected digits over $P _ { i }$ and $Q _ { i }$ , i.e., $\textstyle \sum _ { j = 1 } ^ { 4 } P _ { i , j } G ( z _ { i , j } )$ and $\textstyle \sum _ { j = 1 } ^ { 4 } Q _ { i , j } G ( z _ { i , j + 4 } )$ , and reconstruct the original input mixture (see Fig.5). As described above, we impose the continuous relaxation of the cardinality and All-Different constraints to reason about the the Sudoku structure among cells of the overlapping Sudokus. To demonstrate the power of reasoning, we compared our unsupervised DRNets with supervised start-of-the-art MNIST de-mixing models – CapsuleNet (Sabour et al., 2017) and ResNet (He et al., 2016), and a variant of DRNets that removes the reasoning modules (”DRNets w/o Reasoning”). To saturate the performance of baseline models, we also applied a post-process local search for them to incorporate the Sudoku Rules. Specifically, we did a local search for the top-2 (top-3 would take too long to search) most likely choice of digits for each Sudoku of the two overlapping Sudokus and try to satisfy Sudoku rules with minimal modification compared with the original prediction. We evaluate both the percentage of digits that are correctly de-mixed (digit accuracy) and the percentage of overlapping Sudokus that have all digits correctly de-mixed (Sudoku accuracy). Empowered by reasoning, DRNets significantly outperformed CapsuleNet, ResNet, and DRNets without reasoning, perfectly recovered all digits with the restart mechanism (see Fig.5), and additionally reconstructed the mixture with high-quality (see Fig.1). Moreover, because DRNets solve all instances together (see Algorithm 2), not only can DRNets solve instances directly on the test set from random initialization, DRNets can also generalize from the training set to test set, given enough training examples. DRNets learn to generalize its de-mixing performance on the test set by solving the training set instances self-supervised (Jing & Tian, 2019) by Sudoku rules, instead of labels, and even outperform CapsuleNet and ResNet (Fig.5). Note that, for unseen instances in the test set, we further optimize the instances for 25 steps to achieve the reported performance (Additional details in the supplementary material).
+
+![](images/3619214d930321c561caf7f8744d81d7ecaf8fba6f78df59051aa43145ced933.jpg)  
+Figure 6: The latent space of DRNets for Crystal-Structure-Phase-Mapping. $M$ denotes the number of possible phases. (For Al-Li-Fe, $M = 1 5 9$ ; For Bi- $\mathbf { \mathrm { C u } }$ -V, $M = 1 0 0 .$ .)
+
+Crystal-Structure-Phase-Mapping concerns inferring crystal structures from a set of X-ray diffraction measurements (XRDs) of a given chemical system, satisfying thermodynamic constraints. Crystal structure phase mapping is a very challenging task, a major bottleneck in high-throughput materials discovery: Each X-ray measurement may involve several mixed crystal structures; each chemical system includes hundreds of possible crystal structures; for each crystal structure pattern, we only have a theoretical (idealized) model of pure crystal phases; the thermodynamic rules are also complex; and the crystal patterns are difficult for human experts to interpret. Herein, we illustrate DRNet for crystal structure phase mapping for two chemical systems: (1) a ternary Al-Li-Fe oxide system (Le Bras et al., 2014), which is theoretically based, synthetically generated, with ground truth solutions, and (2) a ternary Bi-Cu-V oxide system, which is a more challenging real experiment-based system, more noisy and uncertain. For each system, each input data point is the XRD of a mixture of crystal structures. Additionally, the input includes the composition graph specifying elemental compositions and the constraint graph of the data points. We also collected a library of possible crystal structures from the International Centre for Diffraction Data (ICDD) database. Each crystal structure (also named phase) is given as a list of diffraction peak location-amplitude pairs, (referred to as stick pattern), representing the ideal phase patterns measured in a perfect condition (see Fig.6). To model more realistic conditions, DRNets simulate the real phase patterns from stick patterns using Gaussian mixture models, where the relative peak locations and mixture coefficients are given by the stick locations and amplitudes. Moreover, the peak width, peak location shift, and peak amplitude variance are parameterized by the latent encoding $z _ { i , j }$ and used by the generative decoder to generate the corresponding possible phase patterns in the reconstructed XRD measurement.
+
+![](images/3b754650d7cdc15fa3366dc63332c76346b78d3b1f58e03094832d0724b9cf5c.jpg)  
+Figure 7: Left: Comparison of phase concentration and reconstruction loss for different methods in Al-Li-Fe oxide system. Note that, 6 pure phases (out of 159 possible candidates) appear in the system and result in 15 different mixtures. Each dot represents an XRD measurement whose size is proportional to the estimated phase concentration. DRNet’s phase concentration closely match the ground truth in contrast to IAFD’s and NMF-k’s. The heatmap on the right shows that DRNets reconstruct the XRD measurements much better than other methods with respect to the L1 loss. Right: DRNets outperform both IAFD and NMF- $\mathbf { \nabla } \cdot \mathbf { k }$ with better reconstruction error and perfect rule satisfaction on both systems. (additional details for Bi-Cu-V in the supplementary material).
+
+<table><tr><td rowspan=1 colspan=1>ChemicalSystems:</td><td rowspan=1 colspan=2>ReconstructionLosses</td><td rowspan=1 colspan=1>PhaseFidelity Loss</td><td rowspan=1 colspan=3>Thermodynamic Rules Satisfaction(Percentage of data points / phasefield that satisfy each constraint)</td></tr><tr><td rowspan=1 colspan=1>Al-Li-Fe</td><td rowspan=1 colspan=1>L1Loss</td><td rowspan=1 colspan=1>L2Loss</td><td rowspan=1 colspan=1>JS distance（×10-2）</td><td rowspan=1 colspan=1>Gibbs</td><td rowspan=1 colspan=1>Gibbs-Alloy</td><td rowspan=1 colspan=1>Phase FieldConnectivity</td></tr><tr><td rowspan=1 colspan=1>DRNets</td><td rowspan=1 colspan=1>0.039</td><td rowspan=1 colspan=1>&lt;0.001</td><td rowspan=1 colspan=1>&lt;0.001</td><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>100%</td></tr><tr><td rowspan=1 colspan=1>IAFD</td><td rowspan=1 colspan=1>5.994</td><td rowspan=1 colspan=1>0.535</td><td rowspan=1 colspan=1>11.30</td><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>100%</td></tr><tr><td rowspan=1 colspan=1>NMF-k</td><td rowspan=1 colspan=1>7.267</td><td rowspan=1 colspan=1>0.438</td><td rowspan=1 colspan=1>56.10</td><td rowspan=1 colspan=1>94%</td><td rowspan=1 colspan=1>87%</td><td rowspan=1 colspan=1>71%</td></tr><tr><td rowspan=1 colspan=1>Bi-Cu-V</td><td rowspan=1 colspan=1>L1Loss</td><td rowspan=1 colspan=1>L2Loss</td><td rowspan=1 colspan=1>JS distance(×10-2)</td><td rowspan=1 colspan=1>Gibbs</td><td rowspan=1 colspan=1>Gibbs-Alloy</td><td rowspan=1 colspan=1>Phase FieldConnectivity</td></tr><tr><td rowspan=1 colspan=1>DRNets</td><td rowspan=1 colspan=1>3.993</td><td rowspan=1 colspan=1>0.196</td><td rowspan=1 colspan=1>8.370</td><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>100%</td></tr><tr><td rowspan=1 colspan=1>IAFD</td><td rowspan=1 colspan=1>7.425</td><td rowspan=1 colspan=1>0.545</td><td rowspan=1 colspan=1>93.36</td><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>99%</td><td rowspan=1 colspan=1>95%</td></tr><tr><td rowspan=1 colspan=1>NMF-k</td><td rowspan=1 colspan=1>8.033</td><td rowspan=1 colspan=1>0.675</td><td rowspan=1 colspan=1>92.63</td><td rowspan=1 colspan=1>51%</td><td rowspan=1 colspan=1>35%</td><td rowspan=1 colspan=1>83%</td></tr></table>
+
+We compared DRNets with IAFD (Bai et al., 2017) and NMF- $\mathbf { \nabla } \cdot \mathbf { k }$ (Stanev et al., 2018), which are both state-of-the-art non-negative matrix factorization (NMF) based unsupervised de-mixing models. NMF-k improves the pure NMF algorithm (Long et al., 2009) by clustering common phase patterns from thousands of runs. However, NMF-k does not directly enforce thermodynamic rules and therefore the solutions produced are often not completely physically meaningful. IAFD uses external mixed-integer programming modules to enforce thermodynamic rules during the de-mixing. However, due to the gap between the external optimizer and NMF module, the solution of IAFD is still far from the ground truth. Our evaluation criteria include reconstruction losses, phase fidelity loss and the satisfaction of thermodynamic rules. Note that, the phase fidelity loss measures the JS distance between the de-mixed phases and the closest ideal phases by fitting the de-mixed phases with the ICDD stick patterns using the physical model (Le Bras et al., 2014). As shown in Fig.7, for the Al-Li-Fe oxide system, the phase concentration (the distribution of de-mixed pure phases over all data points of that chemical system) of either IAFD or NMF-k is far from the ground truth. In contrast, DRNet almost exactly recovered the ground truth solution by seamlessly integrating pattern recognition, reasoning and prior knowledge. Moreover, by explicitly incorporating the ICDD stick pattern information into DRNets, the phases de-mixed by DRNets are much more real than those from IAFD and NMF-k (see phase fidelity loss). For Bi-Cu-V oxide system, DRNets solved this previously unsolved real system, producing valid crystal structures and significantly outperforming IAFD and NMF-k w.r.t. reconstruction errors and phase fidelity loss. In addition, materials science experts thoroughly checked DRNet’s solution of Bi-Cu-V oxide system, approved it, and subsequently discovered a new material that is important for solar fuels technology.
+
+# 5 CONCLUSIONS AND FUTURE WORK
+
+We propose DRNets, a powerful end-to-end framework that combines deep learning with logical and constraint reasoning for solving unsupervised pattern de-mixing tasks. DRNets outperform the state of the art for de-mixing MNIST Sudokus and crystal-structure phase mapping, solving previously unsolved chemical systems substantially beyond the reach of other methods and materials science experts’ capabilities. While we illustrate the potential of DRNets with unsupervised settings, it is straightforward to impose supervision into DRNets. Future research includes exploring DRNets for incorporating other types of constraints, prior knowledge, and objective functions, for other applications.
+
+# REFERENCES
+
+Saeed Amizadeh, Sergiy Matusevych, and Markus Weimer. Pdp: A general neural framework for learning constraint satisfaction solvers. arXiv preprint arXiv:1903.01969, 2019.
+
+Junwen Bai, Johan Bjorck, Yexiang Xue, Santosh K Suram, John Gregoire, and Carla Gomes. Relaxation methods for constrained matrix factorization problems: solving the phase mapping problem in materials discovery. In International Conference on AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, pp. 104–112. Springer, 2017.
+
+Junwen Bai, Sebastian Ament, Guillaume Perez, John Gregoire, and Carla Gomes. An efficient relaxed projection method for constrained non-negative matrix factorization with application to the phase-mapping problem in materials science. In International Conference on the Integration of Constraint Programming, Artificial Intelligence, and Operations Research, pp. 52–62. Springer, 2018.
+
+Stefano Ermon, Ronan Le Bras, Santosh K Suram, John M Gregoire, Carla P Gomes, Bart Selman, and Robert B Van Dover. Pattern decomposition with complex combinatorial constraints: Application to materials discovery. In Twenty-Ninth AAAI Conference on Artificial Intelligence, 2015.
+
+Kuzman Ganchev, Jennifer Gillenwater, Ben Taskar, et al. Posterior regularization for structured latent variable models. Journal of Machine Learning Research, 11(Jul):2001–2049, 2010.
+
+Artur d’Avila Garcez, Marco Gori, Luis C Lamb, Luciano Serafini, Michael Spranger, and Son N Tran. Neural-symbolic computing: An effective methodology for principled integration of machine learning and reasoning. arXiv preprint arXiv:1905.06088, 2019.
+
+Carla P Gomes, Bart Selman, Henry Kautz, et al. Boosting combinatorial search through randomization. AAAI/IAAI, 98:431–437, 1998.
+
+Gordon Royle. Minimum sudoku, 2014. URL http://staffhome.ecm.uwa.edu.au/ ˜00013890/sudokumin.php.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Geoffrey E Hinton, Zoubin Ghahramani, and Yee Whye Teh. Learning to parse images. In Advances in neural information processing systems, pp. 463–469, 2000.
+
+Zhiting Hu, Xuezhe Ma, Zhengzhong Liu, Eduard Hovy, and Eric Xing. Harnessing deep neural networks with logic rules. arXiv preprint arXiv:1603.06318, 2016a.
+
+Zhiting Hu, Zichao Yang, Ruslan Salakhutdinov, and Eric Xing. Deep neural networks with massive learned knowledge. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, pp. 1670–1679, 2016b.
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International Conference on Machine Learning, pp. 448–456, 2015.
+
+Longlong Jing and Yingli Tian. Self-supervised visual feature learning with deep neural networks: A survey. arXiv preprint arXiv:1902.06162, 2019.
+
+Khanrc. Mnist!: learn mnist classifier with very high accuracy, 2017. URL https://github. com/khanrc/mnist.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Laodar. A tensorflow implementation for capsnet, 2017. URL https://github.com/laodar/ tf_CapsNet.
+
+Ronan Le Bras, Richard Bernstein, John M Gregoire, Santosh K Suram, Carla P Gomes, Bart Selman, and R Bruce Van Dover. Challenges in materials discovery–synthetic generator and real datasets. In Twenty-Eighth AAAI Conference on Artificial Intelligence, 2014.
+
+Yann LeCun, Leon Bottou, Yoshua Bengio, Patrick Haffner, et al. Gradient-based learning applied to ´ document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Erik Linder-Noren. Pytorch-gan, 2019. URL https://github.com/eriklindernoren/ PyTorch-GAN.
+
+CJ Long, D Bunker, X Li, VL Karen, and I Takeuchi. Rapid identification of structural phases in combinatorial thin-film libraries using x-ray diffraction and non-negative matrix factorization. Review of Scientific Instruments, 80(10):103902, 2009.
+
+Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. arXiv preprint arXiv:1411.1784, 2014.
+
+David Mitchell, Bart Selman, and Hector Levesque. Hard and easy distributions of sat problems. In AAAI, volume 92, pp. 459–465, 1992.
+
+Rasmus Berg Palm, Ulrich Paquet, and Ole Winther. Recurrent relational networks for complex relational reasoning. arXiv preprint arXiv:1711.08028, 2017.
+
+Deepak Pathak, Philipp Krahenbuhl, and Trevor Darrell. Constrained convolutional neural networks for weakly supervised segmentation. In Proceedings of the IEEE international conference on computer vision, pp. 1796–1804, 2015.
+
+Herbert Robbins and Sutton Monro. A stochastic approximation method. In Herbert Robbins Selected Papers, pp. 102–109. Springer, 1985.
+
+Sara Sabour, Nicholas Frosst, and Geoffrey E Hinton. Dynamic routing between capsules. In Advances in neural information processing systems, pp. 3856–3866, 2017.
+
+Daniel Selsam, Matthew Lamm, Benedikt Bunz, Percy Liang, Leonardo de Moura, and David L Dill. ¨ Learning a sat solver from single-bit supervision. arXiv preprint arXiv:1802.03685, 2018.
+
+Radhika Shivhare and Ch Aswani Kumar. On the cognitive process of abstraction. Procedia Computer Science, 89:243–252, 2016.
+
+Valentin Stanev, Velimir V Vesselinov, A Gilad Kusne, Graham Antoszewski, Ichiro Takeuchi, and Boian S Alexandrov. Unsupervised phase mapping of $\mathbf { X }$ -ray diffraction data by nonnegative matrix factorization integrated with custom clustering. npj Computational Materials, 4(1):43, 2018.
+
+Po-Wei Wang, Priya L Donti, Bryan Wilder, and Zico Kolter. Satnet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver. arXiv preprint arXiv:1905.12149, 2019.
+
+Bing Xu, Naiyan Wang, Tianqi Chen, and Mu Li. Empirical evaluation of rectified activations in convolutional network. arXiv preprint arXiv:1505.00853, 2015.
+
+Jingyi Xu, Zilu Zhang, Tal Friedman, Yitao Liang, and Guy Van den Broeck. A semantic loss function for deep learning with symbolic knowledge. arXiv preprint arXiv:1711.11157, 2017.
+
+Yexiang Xue, Junwen Bai, Ronan Le Bras, Brendan Rappazzo, Richard Bernstein, Johan Bjorck, Liane Longpre, Santosh K Suram, Robert B van Dover, John Gregoire, et al. Phase-mapper: an ai platform to accelerate high throughput materials discovery. In Twenty-Ninth IAAI Conference, 2017.
+
+Jiaxuan You, Rex Ying, Xiang Ren, William L Hamilton, and Jure Leskovec. Graphrnn: Generating realistic graphs with deep auto-regressive models. arXiv preprint arXiv:1802.08773, 2018.
+
+Ning Zhang, Junchi Yan, and Yuchen Zhou. Weakly supervised audio source separation via spectrum energy preserved wasserstein learning. arXiv preprint arXiv:1711.04121, 2017.
+
+Xingyi Zhou, Xiao Sun, Wei Zhang, Shuang Liang, and Yichen Wei. Deep kinematic pose regression. In European Conference on Computer Vision, pp. 186–201. Springer, 2016.
+
+Xingyi Zhou, Qixing Huang, Xiao Sun, Xiangyang Xue, and Yichen Wei. Weaklysupervised transfer for 3d human pose estimation in the wild. In IEEE International Conference on Computer Vision, volume 206, pp. 3, 2017.
+
+# A SUPPLEMENTARY MATERIALS
+
+Herein, we provide additional details about DRNets and our experimental settings for a better understanding of DRNets and reproducibility of our results. Code and datasets to reproduce the experiments will be provided with the final version of the paper.
+
+# A.1 CONTINUOUS RELAXATION
+
+In this section, we provide more relxations for other constraints such as SAT constraints and provide an intuitive high-level informal proof that all the relaxations converge to a valid solution of the discrete version when it achieves its minimal value. Fig.8 summarizes the relaxations.
+
+Figure 8: Examples of continuous relaxations: $e _ { i , j } , P _ { i } , Q _ { i } , P _ { M } , N _ { c } , N _ { l } , K _ { j } , \lambda _ { h } ,$ $B _ { i }$ denote binary variables, the discrete distribution over digits 1 to 4, the discrete distribution over digits 5 to 8, the discrete distribution over values 1 to $M$ , the number of clauses, the number of literals, the number of literals in the $j$ -th clause, the weights of entropy terms, and the Bernoulli distribution for the $i$ -th literal. ”leaky relu” is the leaky ReLU.   
+
+<table><tr><td rowspan=1 colspan=1>Cardinality Constraintei,j ∈{0,1} j= 1...8s.t.∑=1eij=1and∑j=5eij=1</td><td rowspan=1 colspan=1>Cardinality ConstraintRelaxationmin H(Pi)+H(Qi)e=-∑=1Pi,jlog Pi,j-∑=1Qi,jlogQi,j</td></tr><tr><td rowspan=1 colspan=1>All-Different ConstraintFor all constrained set Ss.t.∑i∈s ei,j = 1 forj = 1...8</td><td rowspan=1 colspan=1>All-Different ConstraintRelaxationFor all constrained set SmaxH(Ps)+H(Qs)0</td></tr><tr><td rowspan=1 colspan=1>k-Sparsity Constrainteij ∈{0,1} j=1...M s.t.∑1ei,j ≤k</td><td rowspan=1 colspan=1>k-Sparsity Constraint Relaxationmin max{H(Pm),c}，where c &lt;log k0</td></tr><tr><td rowspan=1 colspan=1>Integer Programming Encoding of SATFor any literal xi and its negation xi,s.t.xi,xi∈{0,1} and xi+xi=1For any clause Cj = αj,1 V.V ajKj</td><td rowspan=1 colspan=1>SATRelaxationFor any literal xi and its negation xi (i = 1..Nt),we modela distribution Bi~Bern(piqi)，s.t.xi=Pi,and xi=qiFor all clause Cj = αj,1 V …V aj,Kj,j= 1 ... NcmjnΣ1leakyrelu(1-Σaj）+n∑1H(B）</td></tr></table>
+
+For cardinality constraints, when the entropy of distribution $P _ { i }$ and $Q _ { i }$ reaches 0, all the probability mass collapses to only one variable. Therefore, all $P _ { i , j }$ and $Q _ { i , j }$ are either 0 or 1, which is a valid solution of the original discrete constraints.
+
+For All-Different constraints, we maximize the entropy of the averaged digit distribution for all cells in a constrained set $S$ , i.e., $H ( \bar { P } _ { S } )$ . Note that, the All-Different constraints are imposed together with the cardinality constraints. Therefore, when the entropy of the digit distribution in each cell is zero, we know that the digit distribution of each cell converges to one digit. Hence, if $H ( \bar { P } _ { S } )$ reaches its maximum, i.e., $\log | S |$ , we have $\begin{array} { r } { \frac { 1 } { | S | } \sum _ { i \in S } P _ { i , j } = \frac { 1 } { | S | } } \end{array}$ for all digit $j$ . Crossed with the fact that $P _ { i , j }$ are either 0 or 1 when the cardinality constraints are satisfied, we know that only one $P _ { i , j }$ is equal to 1 for all cell $i$ in the set $S$ and others are 0, which directly states the All-Different constraints.
+
+We derive the k-Sparsity constraints in a similar way as the cardinality constraints except that we now want to force the distribution to concentrate on at most $\mathrm { k }$ digits. By normalizing the values of discrete variables $e _ { i , j }$ $( j = 1 . . . M )$ to a discrete distribution $P _ { M }$ , we can minimize the entropy of distribution $P _ { M }$ to at most $\log k$ , which is the maximal entropy when the distribution concentrates on only $k$ values. Though, $H ( P _ { M } ) < \log k$ is not a sufficient condition for $\mathbf { k }$ -sparsity, we can initialize the threshold $c$ of $\mathbf { k }$ -sparsity constraints to $\log k$ and dynamically adjust the value of $c$ based on the satisfaction of the $\mathbf { k }$ -sparsity constraints. In practice, it works well with the supervision from other modules, such as the self-reconstruction.
+
+For SAT constraint relaxations, the key idea is to minimize the entropy of the Bernoulli distribution over each literal to force it converge to either 1 or 0. Then, we maximize the sum of the value of literals in each clause (or their negation) to encourage one of the literals to be 1. However, maximizing the sum of the value of literals does not necessarily give you a valid assignment because there could exist an assignment that the sum of literals in some clauses are 0 and the sum of literals in other clauses are very large. Therefore, we use leaky rule ( $\mathrm { { X u } }$ et al., 2015) function to discount the loss when the sum is larger than 1. As shown in Fig.8, we formulate the relaxation loss function in a form to be minimized. For k-SAT problems with $N _ { c }$ clauses, we can set the leaky ratio to be 1N k , so that any invalid assignment cannot have a loss that is less or equal to 0. On the other hand, for any valid assignment, the sum of literals in each clause is at least 1. Thus, we can obtain a valid assignment of k-SAT constraints by minimizing the loss function to 0.
+
+We describe other task specific constraints (e.g., phase field connectivity constraints) in the following experimental sections.
+
+# A.2 CONSTRAINT-AWARE STOCHASTIC GRADIENT DESCENT:
+
+Algorithm 2 Constraint-aware stochastic gradient descent optimization of deep reasoning networks.
+
+Input: (i) Data points $\{ x _ { i } \} _ { i = 1 } ^ { N }$ . (ii) Constraint graph. (iii) Penalty functions $\psi ^ { l } ( \cdot )$ and $\psi _ { j } ^ { g } ( \cdot )$ for the local and the global constraints. (iv) Pre-trained or parametric generative decoder $G ( \cdot )$ .   
+1: Initialize the penalty weights $\lambda ^ { l } , \lambda _ { j } ^ { g }$ and thresholds for all constraints.   
+2: for number of optimization iterations do   
+3: Batch data points $\{ \mathbf { x } _ { 1 } , . . . , \mathbf { x } _ { m } \}$ from the sampled (maximal) connected components.   
+4: Collect the global penalty functions $\{ \psi _ { j } ^ { g } ( \cdot ) \} _ { j = 1 } ^ { \hat { M } }$ concerning those data points.   
+5: Compute the latent space $\{ \phi _ { \theta } ( \mathbf { x } _ { 1 } ) , . . . , \phi _ { \theta } ( \mathbf { x } _ { m } ) \}$ from the encoder.   
+6: Adjust the penalty weights $\lambda _ { l } , \lambda _ { j } ^ { g }$ and thresholds accordingly.   
+7: minimize $\begin{array} { r } { \frac { 1 } { m } \big ( \sum _ { i = 1 } ^ { m } \mathcal { L } ( G ( \phi _ { \theta } ( \mathbf { x } _ { i } ) ) , \mathbf { x } _ { i } ) + \lambda _ { l } \psi ^ { l } ( \phi _ { \theta } ( \mathbf { x } _ { i } ) ) \big ) + \sum _ { j = 1 } ^ { M } \lambda _ { j } ^ { g } \psi _ { j } ^ { g } ( \{ \phi _ { \theta } ( \mathbf { x } _ { k } ) | k \ \in \ S _ { j } \} ) } \end{array}$ using any standard gradient-based optimization method and update the parameters $\theta$ .
+
+We introduce a variant of standard SGD method called constraint-aware SGD, which is conceptually similar to the optimization process in GraphRNN (You et al., 2018), to tackle the optimization of global penalty functions $\psi _ { j } ^ { \dot { g _ { ( } } } ( \{ \phi _ { \theta } ( \mathbf { x } _ { k } ) | k \in \mathsf { \bar { S } } _ { j } \} )$ , which involve several (potentially all) data points. We define a constraint graph, an undirected graph in which each data point forms a vertex and two data points are linked if they are in the same global constraint. Constraint-aware SGD batches data points from the randomly sampled (maximal) connected components in the constraint graph, and optimizes the objective function w.r.t. the subset of global constraints concerning those data points and the associated local constraints. For example, in Multi-MNIST-Sudoku, each overlapping Sudoku forms a maximal connected component, we batch the data points from several randomly sampled overlapping Sudokus and optimize the All-Different constraints (global) as well as the cardinality constraints (local) within them. However, in Crystal-Structure-Phase-Mapping, the maximal connected component becomes too large to batch together, due to the constraints (phase field connectivity and Gibbs-alloying rule) concerning all data points in the composition graph. Thus, we instead only batch a subset (still a connected component) of the maximal connected component – e.g., a path in the composition graph, and optimize the objective function that only concerns constraints within the subset (along the path). By iteratively solving sampled local structures of the ”large” maximal component, we cost-efficiently approximate the entire global constraint.
+
+Moreover, for optimizing the overall objective, constraint-aware SGD dynamically adjusts the thresholds and the weights of constraints according to their satisfiability, which can involve nondifferentiable functions. Specifically, we initialize penalty weights of constraints and thresholds for penalty functions using hyper-parameters. During training, we check the satisfiability of constraints (this step could involve non-differentiable functions) after several epochs and increase the penalty for violated constraints. For example, the threshold $c$ of $\mathbf { k }$ -sparsity is initialized as $\log k$ , which is the entropy of the case that the probability mass is evenly distributed among $k$ entities. Thus, it could be the case that there are more than $k$ entities, but their probability mass is not evenly distributed. Hence, we check the satisfiability of k-sparsity constraint: if the entropy is already below the current threshold $( \log k )$ and there are still more than $k$ entities with probability mass more than $\epsilon \left( 0 . 0 1 \right)$ , we decrease the threshold $c$ to keep enforcing the model to minimize the entropy to reach the k-sparsity.
+
+Finally, to better exploit parallelization, DRNets solve all instances together using constraint-aware SGD (see Algorithm 2).
+
+# A.3 RESTART MECHANISM FOR DRNETS:
+
+Note that, since DRNets are an unsupervised framework, we can apply the restart (Gomes et al., 1998) mechanism, i.e., we can re-run DRNets for unsolved instances. Specifically, since DRNets directly incorporate logical constraints, we can check whether those constraints are satisfied at the end of a run. If not, for instances with violated constraints, we re-run the algorithm again on them. We only applied restart mechanism on Multi-MNIST-Sudoku and other NP-C problems (in the appendix) such as 3-SAT problems and standard Sudoku problems. For crystal-structure phase mapping, the results generated from one run of DRNets is already good enough.
+
+# A.4 EXPERIMENTAL CONFIGURATION
+
+All the experiments are performed on one NVIDIA Tesla V100 GPU with 16GB memory. For the training process of our DRNets, we select a learning rate in $\{ 0 . 0 0 0 1 , 0 . 0 0 0 5 , 0 . 0 0 1 \}$ with Adam optimizer (Kingma & Ba, 2014), for all the experiments.
+
+For baseline models, we followed their original configurations and further fine-tuned their hyperparameters to saturate their performance on our tasks.
+
+# A.4.1 MULTI-MNIST-SUDOKU
+
+For Multi-MNIST-Sudoku, we compared DRNets with CapsuleNet (Sabour et al., 2017) and ResNet (He et al., 2016). Because Sabour et al. (2017) did not provide a source code for CapsuleNet, we adopted the implementation of Laodar (2017), with minor modifications. For ResNet, we adopted a 18-layer ResNet architecture (Khanrc, 2017) to saturate its performance.
+
+In Multi-MNIST-Sudoku, a data point corresponds to a $3 2 \times 3 2$ image of overlapping digits. For the optimization mode of DRNets, we generated 160, 000 input data points that all come from the test set of MNIST (LeCun et al., 1998) and every 16 data points form a 4-by-4 overlapping Sudokus. Thus, these 160, 000 data points form 10, 000 Sudokus. These 10, 000 Sudokus are used as the test set and shared across DRNets and baselines. For the generalization mode of DRNets, we split the training set of MNIST into three parts: 160, 000 data points for DRNets learning, 25, 000 original MNIST images for training conditional GAN and another 160, 000 data points for validation. Note that these three datasets are disjoint. Baselines share the same training set as the generalization mode of DRNets. By using the constraint-aware SGD, DRNet batches every 16 data points together, which forms an overlapping Sudoku as well as a maximal connected component in the constraint graph, to enforce the All-Different constraints among the cells of each Sudoku.
+
+DRNet for Multi-MNIST-Sudoku: the encoder is made of two ResNet-18 models adapted from the PyTorch source code. The output layer for the first network has 8 dimensions, which models the two distributions $P _ { i }$ and $Q _ { i }$ for the two overlapping digits. Another network outputs eight 100-dimensional (800 dimensions in total) latent encoding $z _ { i , j }$ to encode the shape of the possible eight digits conditioned on the input mixture, and is used by the generative decoder to generate the reconstructed digits. We use a conditional GAN (Mirza & Osindero, 2014) as our generative decoder, which is pre-trained using the digits in the partial training set (see the paragraph above) of MNIST. Note that this is the only supervision we have in this task, which is even weaker than the general concept of the weakly-supervised setting (Zhang et al., 2017). We adopted the implementation of Linder-Noren (2019) for our conditional GAN. On the other hand, the 10,000 overlapping Sudokus in the test set were all generated using the digits in the test set of MNIST, which had never been seen, even by the conditional GAN. Moreover, we overlap the images of two digits pixel-wisely, maximizing the whiteness of the two images. For robustness concern, we used $L 1$ loss as the reconstruction loss between the reconstructed mixture and the original input. For the initial weights, we set 0.01 for the cardinality constraints, 1.0 for the All-Different constraints, and 0.001 for the $L 1$ loss. Finally, we trained DRNets for 100 epochs with a batch size of 100, and it took 50 minutes to finish the optimization and achieve the reported performance for the 10,000 overlapping Sudokus.
+
+Figure 9: Accuracy comparison. We show ”test time $^ +$ training time” for supervised baselines and the generalization mode of DRNet, and ”solving time” for the optimization mode of DRNets.   
+
+<table><tr><td rowspan=2 colspan=1>Method</td><td rowspan=1 colspan=2>Accuracy (%)</td><td rowspan=2 colspan=1>Time</td></tr><tr><td rowspan=1 colspan=1>Digit</td><td rowspan=1 colspan=1>Sudoku</td></tr><tr><td rowspan=1 colspan=1>DRNets (Optimization w/Restart)</td><td rowspan=1 colspan=1>100.0</td><td rowspan=1 colspan=1>100.0</td><td rowspan=1 colspan=1>50min</td></tr><tr><td rowspan=1 colspan=1>DRNets (Optimization)</td><td rowspan=1 colspan=1>99.9</td><td rowspan=1 colspan=1>98.6</td><td rowspan=1 colspan=1>28min</td></tr><tr><td rowspan=1 colspan=1>DRNets (Optimization W/oReasoning)</td><td rowspan=1 colspan=1>88.8</td><td rowspan=1 colspan=1>15.0</td><td rowspan=1 colspan=1>110min</td></tr><tr><td rowspan=1 colspan=1>DRNets (Generalization)</td><td rowspan=1 colspan=1>98.0</td><td rowspan=1 colspan=1>75.7</td><td rowspan=1 colspan=1>13min+4hrs</td></tr><tr><td rowspan=1 colspan=1>CapsuleNet</td><td rowspan=1 colspan=1>97.9</td><td rowspan=1 colspan=1>50.9</td><td rowspan=1 colspan=1>1min+30min</td></tr><tr><td rowspan=1 colspan=1>CapsuleNet + local search</td><td rowspan=1 colspan=1>97.9</td><td rowspan=1 colspan=1>57.8</td><td rowspan=1 colspan=1>3hrs+30mins</td></tr><tr><td rowspan=1 colspan=1>ResNet-18</td><td rowspan=1 colspan=1>97.7</td><td rowspan=1 colspan=1>68.5</td><td rowspan=1 colspan=1>3min+10hrs</td></tr><tr><td rowspan=1 colspan=1>ResNet-18 +local search</td><td rowspan=1 colspan=1>97.7</td><td rowspan=1 colspan=1>88.3</td><td rowspan=1 colspan=1>3hrs+10hrs</td></tr></table>
+
+For the generalization mode of DRNets, we first ”train” DRNets on the training set and validate its generalization performance on the validation set to apply the early stop mechanism. Finally, we start from the ”trained” DRNets and further optimize it for 25 steps on the test set to achieve the reported performance. Note that, to generalize well on the test set, we ”trained” DRNets for a longer time than the optimization mode. Essentially, the procedure of the generalization mode of DRNets is similar to standard supervised learning process except that we do not need labels to supervise DRNets. In contrast, DRNets are really ”self-supervised” (Jing & Tian, 2019) by the Sudoku rules and the self-reconstruction, instead of the standard supervision by labeled data. Note that, during the test, instead of predicting the overlapping digits directly as other networks, we further optimize DRNets on the test set for 25 epochs to achieve a better result.
+
+# A.4.2 CRYSTAL-STRUCTURE-PHASE-MAPPING
+
+We illustrate DRNets for crystal structure phase mapping for two chemical systems: (1) a ternary Al-Li-Fe oxide system (Le Bras et al., 2014), which is theoretically-based, synthetically generated, with ground-truth solutions, and (2) a ternary $\mathbf { B i - C u - V }$ oxide system, which is a more challenging real system obtained from chemical experiments and is more noisy and uncertain. For each system, the input data points are mixtures of XRDs, associated with a composition graph identifying elemental compositions and the constraint graph of data points. Specifically, each XRD data point is associated with a 3-dimensional composition vector, which is the proportion of the three different metal elements at that data point (e.g., $[ 8 0 \%$ of Al, $5 \%$ of Fe, $15 \%$ of Li]) and could help identify possible phases. Then, we can locate each data point into a triangular system. Note that, since the vector is a probability distribution, there are only 2 degrees of freedom and we can plot it in a 2-D triangle (See Fig.11). After locating each data point into the 2-D triangle as vertices, we did a Delaunay triangulation over those points to build edges among vertices. Therefore, we can use Breadth-First Search on this graph to sample paths in the composition graph and infer thermodynamic rules accordingly.
+
+The XRD pattern of each data point is a $D$ -dimensional vector representing the intensity of the mixture of XRDs at different diffraction angles (referred as $Q$ values). For Al-Li-Fe oxide system, we have 231 data points (mixtures of XRDs) in the composition graph, 159 stick patterns for the possible phases, and each data point has 650 different $\mathrm { Q }$ values $Q _ { i } \in [ 1 5 ^ { \circ } , 8 0 ^ { \circ } ]$ and the corresponding intensities $I _ { i } \in [ 0 , 1 ]$ . For $\mathrm { B i - C u - V }$ oxide system, we have 353 data points in the composition graph, 100 stick patterns for the possible phases, and each data point has 4096 different $\mathrm { Q }$ values $Q _ { i } \in [ 5 ^ { \circ } , 4 5 ^ { \circ } ]$ and the corresponding intensities $I _ { i } \in [ 0 , 1 ]$ . To better utilize the memory, we down-sampled the raw data of $\mathrm { B i - C u - V }$ oxide system to 512 different $\mathrm { Q }$ values. Note that, though we have hundreds of possible pure phases for each system, only a few phases would appear. For example, in Al-Fe-Li oxide system, only 6 of them appear and there are 15 different mixtures of those 6 pure phases exist in this system. For the Bi-Cu-V-O system, there are 13 pure phases and 19 different mixtures. Note that, each XRD data point is like a cell in the Multi-MNIST-Sudoku (with mixed pure phases) and each pure phase is like a digit. For Multi-MNIST-Sudoku, we know a priori that there are exact 2 digits in each cell but the number of mixed pure phases in each XRD is undetermined (1 to 3). Moreover, the number of possible candidate phases is way more than possible digits (e.g., 159 vs 8), which is the reason why this task is so challenging.
+
+![](images/6f952a058d2efeb27f80dc10596d52654a08e6c0dba7ae0f64f82c15728d267e.jpg)  
+Figure 10: Deep reasoning networks (DRNets) for crystal-structure-phase-mapping. (a) Prior knowledge includes the ICDD stick patterns of possible pure phases, which are used to build the GMM generative module in the decoder, and the thermodynamic rules that help DRNets reason about the mixture of XRD patterns. (b) reasoning modules batch data points involved in a connected component of the constraint graph (a path in the composition graph) together, enforce that the structure of the latent space satisfies prior knowledge, and dynamically adjust the weights of constraints based on their satisfiability. (c) The overall objective combines responses from the generative decoder and the reasoning modules.
+
+![](images/38a8ea9f2f019030a29a39e68648114e1ff61f78ba138adefcc3aa789eb9fe2e.jpg)  
+Figure 11: The composition graph of the Al-Fe-Li oxide system. The red path is a sampled path in the composition graph.
+
+We also collected a library of possible crystal structures from the International Centre for Diffraction Data (ICDD) database. Each crystal structure (also named phase) is given as a list of diffraction peak location-amplitude pairs, (referred to as stick pattern), representing the ideal phase patterns measured in a perfect condition (see Fig.12). To model more realistic conditions, DRNets simulate the real phase patterns from stick patterns using Gaussian mixture models, where the relative peak locations and mixture coefficients are given by the stick locations and amplitudes. Moreover, the peak width, multiplicative location shift, and possible amplitude variance are parameterized by the latent encoding $z _ { i , j }$ and used by the generative decoder to generate the corresponding possible phase patterns in the reconstructed XRD measurement.
+
+Imposing thermodynamic rules is challenging, especially when constraints, such as phase field connectivity and Gibbs-alloying rule, potentially concern all data points in the composition graph.
+
+![](images/82af903cba2a3a578c7e94d4eaaf09615f726588fb54c60441fcf611d2562489.jpg)  
+Figure 12: Some examples of stick patterns and their corresponding Gaussian Mixture Models. The horizontal axis denotes the Q values, and the vertical axis denotes the diffraction intensity.
+
+In Multi-MNIST-Sudoku, where each overlapping Sudoku naturally forms the maximal connected components in the constraint graph, we can easily batch every 16 data points together to reason about the All-Different constraints among them. However, in Crystal-Structure-Phase-Mapping, since the maximal connected component involves all data points in the composition graph, neither batching all data points into the memory nor reasoning about the whole graph is tractable. Therefore, we devised a strategy of sampling the large connected component through many local structures (still connected components) and iteratively solve each of them. Specifically, for each oxide system, we sampled 100,000 paths in the composition graph via Breadth First Search to construct a path pool. Then, for every iteration, DRNets randomly sample a path from the pool and batches the data points along that path (see 10). Finally, we only reason about the thermodynamic rules along the path. By iteratively solving sampled local structures (paths) of the ”large” maximal component, we can cost-efficiently approximate all global constraints.
+
+We summarize the thermodynamic rules we imposed in DRNets:
+
+Gibbs Phase Rule: This rule states the maximum number of co-existing phases, which is imposed via our relaxation of the $\mathbf { k }$ -sparsity constraints.
+
+Gibbs-Alloying Rule: This rule states that if ”alloying” happens, then the maximum number of possible co-existing phases should decrease by one. ”Alloying” is a phenomenon that the stick locations of a phase (crystal structure) shift (change) along with adjacent data points. DRNet explicitly models the shifting ratio in the generative decoder and penalize the difference between adjacent data points along our sampled path. The reasoning module keeps track of the difference of shifting ratio between adjacent data points, and when it is larger than a threshold (0.001), we confirm the existence of ”alloying” and reduce the maximum number of possible co-existing phases by one via adjusting the threshold $c$ in the $\mathbf { k }$ -Sparsity Constraints.
+
+Phase Field Connectivity: This states that the distribution (also referred as activation) of a phase field should form a connected component in the composition graph, and the variation of the activation of each phase should also be smooth (see Fig.13). (Herein, the phase field refers to the co-existence of a combination of phases, including the existence of a pure phase.) We impose this rule by penalizing the difference of the phase distribution $P _ { i }$ between adjacent data points along our sampled path.
+
+Multiplicative Shifting: This states how a cubic crystal structure shifts when ”alloying” happens, and this can also be used to approximate the shifting of other crystal structures. We explicitly modeled the multiplicative shifting in our generative decoder.
+
+Noise Threshold: To remove negligible activations that are mainly caused by noise we applied simple post-processing that cuts-off all the activations that are lower than $1 . 0 \%$ .
+
+Here, we visualized the DRNets’ solution of Bi-Cu-V oxide system (see Fig.13 and the comparison among different methods Fig.14).
+
+In our comparison, we evaluated the percentage of data points or phase field that satisfy each thermodynamic rule. Though IAFD enforced the thermodynamic rule using an external mixed-integer programming module, it may compromise some rules to achieve a better reconstruction error, which explains IAFD’s result for Bi-Cu-V oxide system. The phase fidelity loss we mentioned in our comparison is the JS distance between the de-mixed pure phase and the closest ideal phase generated using the ICDD stick patterns and the physical model proposed in Le Bras et al. (2014). The reason of using JS distance to measure the fidelity is that the location of peaks are the most important characteristics of a phase pattern. Therefore, by normalizing the XRD patterns of pure phases into probability distributions, we can use the JS distance to measure the mismatch of ”peaks” between them.
+
+![](images/a7f70b37030e8951b3372b70d44caba361d0ff832fbf64cdb7631add8996cf6c.jpg)  
+Figure 13: DRNets’ solution for the Bi-Cu-V oxide system. a. The de-mixed crystal phases for the 353 XRD measurements of the Bi-Cu-V oxide system (each plot includes the signal for the recognized phase and the corresponding ICDD stick pattern). b. DRNets’ phase concentration maps for the corresponding phases on the left of the map. Dot sizes are proportional to their estimated phase concentrations and heatmap denotes estimated shifting (alloying). c. DRNets’ crystal phase map for the Bi-Cu-V-O system in the composition graph; the phase fields are labeled with corresponding crystal phases.
+
+![](images/10b97ec8043db0757adb5a4fa2094bb3cd2621439b5a1ef004b9ff59fa58a6e0.jpg)  
+Figure 14: Comparison of the activation map and the heatmap of L1 reconstruction loss for different methods for the Bi-Cu-V oxide system: Each row denotes the activation of the different phases for the the different methods; Though we do not have ground truth for the $\mathrm { B i - C u - V }$ oxide system, the solution generated by DRNets satisfies all thermodynamic rules with excellent reconstruction performance; The heatmap on the right shows that DRNets reconstruct the XRD measurements much better than other methods with respect to the (log scale) L1 reconstruction under physical constraints of decomposed phases; In addition, materials science experts thoroughly checked DRNets’ solution of Bi-Cu-V oxide system, approved it, and subsequently discovered a new material that is important for solar fuels technology.
+
+In terms of the optimization process, DRNets took about 30 minutes to achieve the reported performance for both systems. IAFD and NMF-k have a similar time performance but a much worse performance w.r.t. the solution quality. In fact, for the Bi-Cu-V oxide system, both NMF-k’s solution and IAFD’s solution are not physically meaningful.
+
+In summary, by combining reasoning and deep learning, DRNets significantly outperformed the state of the art and human experts on the crystal-Structure-Phase-Mapping instances, recovering more precise, interpretable, and physically meaningful crystal structure pattern decompositions, and even solving phase diagrams of chemical systems that had not been solved before, such as the Bi-Cu-V-O system, but also other systems not reported here.
+
+# A.4.3 OTHER EXPERIMENTS FOR COMBINATORIAL PROBLEMS
+
+As a proof of concept of how DRNets can encode general combinatorial constraints using our entropybased continuous relaxation, we solved 9-by-9 Sudoku puzzles and Boolean satisfiability problems (SAT) using DRNets. For those two tasks, we use a 3-layer-fully-connected network as our encoder and the reasoning modules.
+
+![](images/d703bf52e4816449709da7f68ccac338b839581e553917cd4ced878e0921fe3c.jpg)  
+Figure 15: A standard 9-by-9 Sudoku puzzle: a partially filled Soduku has to be completed as a valid Sudoku.
+
+For 9-by-9 Sudoku puzzles, we generated 10,000 instances using the dataset gathered by Gordon Royle (2014), where each Sudoku instance has 24 to 32 (uniformly distributed) known cells and is guaranteed to have one unique solution (e.g., see Fig.15). Because a standard 9x9 Sudoku puzzle requires reasoning about the unknown structure based on given clues, we need to treat each entire Sudoku as a single input data point. Therefore, in this task, even the All-Different constraints are conceptually the local constraints since each of them only concerns a single data point. We used a one-hot encoding for digits 1 to 9 and the empty cell (denoted as 0), and the entire Sudoku is an 810-dimensional input data. We used a 3-layer-fully-connected network with batch normalization (Ioffe & Szegedy, 2015) as the encoder, where every hidden layer has 2048 units and the output is an 81-by-9 matrix, which represents the digit distributions (1 to 9) for 81 cells. Moreover, we enforced the distribution of every known cell to collapse to the digit in that cell. For the initial weights, we set 0.0001 for the cardinality constraints and 1.0 for the All-Different constraints. Finally, we trained DRNets for 800,000 iterations with a batch size of 500, and it took 1 hour to solve the $1 0 , 0 0 0 9 \mathrm { x } 9$ Sudokus with the accuracy reported in this paper.
+
+In our experiments, DRNets achieved the same level of performance as the Recurrent Relational Networks (RRNets) (Palm et al., 2017), which is the state-of-the-art supervised deep learning $9 \mathrm { x } 9$ Sudoku solver (see Table 1).
+
+<table><tr><td rowspan=1 colspan=1>Instances (10,000)</td><td rowspan=1 colspan=1>DRNets</td><td rowspan=1 colspan=1>DRNets+Restart</td><td rowspan=1 colspan=1>NeuralSAT</td><td rowspan=1 colspan=1>PDP</td><td rowspan=1 colspan=1>RRNets</td></tr><tr><td rowspan=1 colspan=1>3-SAT n=30 m=129</td><td rowspan=1 colspan=1>81.0% (4min)</td><td rowspan=1 colspan=1>99.0% (33min)</td><td rowspan=1 colspan=1>45.5% (2min+1hr)</td><td rowspan=1 colspan=1>78.9% (5min+2hr)</td><td rowspan=1 colspan=1>NA</td></tr><tr><td rowspan=1 colspan=1>3-SAT n=50 m=215</td><td rowspan=1 colspan=1>63.3% (7min)</td><td rowspan=1 colspan=1>94.0% (47min)</td><td rowspan=1 colspan=1>26.1% (3min+1hr)</td><td rowspan=1 colspan=1>62.2% (8min+2hr)</td><td rowspan=1 colspan=1>NA</td></tr><tr><td rowspan=1 colspan=1>3-SAT n=100 m=430</td><td rowspan=1 colspan=1>34.7% (17min)</td><td rowspan=1 colspan=1>77.9% (2hr)</td><td rowspan=1 colspan=1>4.7% (5min+1hr)</td><td rowspan=1 colspan=1>31.4%(2hr+2hr)</td><td rowspan=1 colspan=1>NA</td></tr><tr><td rowspan=1 colspan=1>3-SAT n=30,m=90</td><td rowspan=1 colspan=1>97.9% (5min)</td><td rowspan=1 colspan=1>99.9% (6min)</td><td rowspan=1 colspan=1>78.5% (2min + 1hr)</td><td rowspan=1 colspan=1>99.1% (4min+ 2hr)</td><td rowspan=1 colspan=1>NA</td></tr><tr><td rowspan=1 colspan=1>3-SAT n=50,m=150</td><td rowspan=1 colspan=1>98.2%(7min)</td><td rowspan=1 colspan=1>99.4% (8min)</td><td rowspan=1 colspan=1>70.1% (3min + 1hr)</td><td rowspan=1 colspan=1>99.2%(7min +2hr)</td><td rowspan=1 colspan=1>NA</td></tr><tr><td rowspan=1 colspan=1>3-SAT n=100,m=300</td><td rowspan=1 colspan=1>98.1% (20min)</td><td rowspan=1 colspan=1>99.7% (22min)</td><td rowspan=1 colspan=1>52.9% (5min + 1hr)</td><td rowspan=1 colspan=1>99.1% (2hr + 2hr)</td><td rowspan=1 colspan=1>NA</td></tr><tr><td rowspan=1 colspan=1>9x9Sudoku</td><td rowspan=1 colspan=1>99.5% (1hr)</td><td rowspan=1 colspan=1>99.8% (1hr)</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>NA</td><td rowspan=1 colspan=1>99.6% (lmin+1day)</td></tr></table>
+
+Table 1: Percentage of instances solved for 3-SAT $( m / n = 4 . 3$ and $m / n = 3 . 0 $ ) and standard $9 \mathrm { x } 9$ Sudoku (24 to 32 known cells). We show the ”test time $^ +$ training time” for supervised baselines and the ”solving time” for our unsupervised DRNets. The units min, hr, day denote minute(s), hour(s) and day(s). $m , n$ denote the number of literals and clauses, respectively. NA, not applicable. DRNets, without supervision, outperform the supervised state of the art.
+
+For SAT problems, we generated 10,000 satisfiable random 3-SAT instances of different difficulties based on the number of literals $n$ and the number of clauses $m$ , and our goal is to find a valid assignment for each literal. We challenged our DRNet with the hardest random 3-SAT instances, where #clauses/#literal ${ = } 4 . 3$ (Mitchell et al., 1992), i.e., $n = 3 0$ , $m = 1 2 9$ , $n = 5 0$ , $m = 2 1 5$ and $n = 1 0 0$ , $m = 4 3 0$ . For easier instances (e.g. #clauses/#literals $= 3 . 0$ ), DRNets can almost solve all instances (see Table 1).
+
+We use a 3-layer-fully-connected network as the encoder, where the number of hidden units in the network is 2048, 2048, 2048. We used the standard CNF representation of 3-SAT as the input data, so that each data point is an $m$ -by-3 matrix and the three values in the $j$ -th row represent the three literals in the $j$ -th clauses. For the initial weights, we select a value from $\{ 0 . 0 5 , 0 . 0 3 , 0 . 0 2 5 , 0 . 0 2$ $0 . 0 1 \}$ to be the weight of the entropy loss as we described in the Fig.4 of the main paper. For the three settings of different difficulty, we consistently trained DRNets with a batch size of 100 and the running time for solving 10,000 instances varies from several minutes to a couple of hours.
+
+We compared DRNets with NeuroSAT Selsam et al. (2018) and PDP (Amizadeh et al., 2019). Both NeuroSAT and PDP are the state-of-the-art deep learning SAT solvers with one-bit supervision. In addition, PDP needs extra optimizing process to solve SAT instances during the test phase, where it also applied the restart mechanism in their framework. For fair comparison, we saturated the performance of all our baseline models. For all instances, DRNets took less than 2 hours to achieve the reported performance with the restart mechanism. Without supervision, DRNets outperformed both supervised baseline models.
+
+Interestingly, though DRNets are best suited for problems that combine deep learning and reasoning, such as de-mixing Multi-MNIST-Sudokus or crystal structure phase mapping, it still achieved such a promising result in pure combinatorial problems. These results further demonstrate that DRNets can encode a broad range of combinatorial constraints and prior knowledge and effectively combine deep learning with reasoning.

parse/train/Hye4KeSYDr/Hye4KeSYDr.md

@@ -0,0 +1,320 @@
+# EVALUATIONS AND METHODS FOR EXPLANATION THROUGH ROBUSTNESS ANALYSIS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Among multiple ways of interpreting a machine learning model, measuring the importance of a set of features tied to a prediction is probably one of the most intuitive way to explain a model. In this paper, we establish the link between a set of features to a prediction with a new evaluation criterion, robustness analysis, which measures the minimum distortion distance of adversarial perturbation. By measuring the tolerance level for an adversarial attack, we can extract a set of features that provides the most robust support for a current prediction, and also can extract a set of features that contrasts the current prediction to a target class by setting a targeted adversarial attack. By applying this methodology to various prediction tasks across multiple domains, we observe the derived explanations are indeed capturing the significant feature set qualitatively and quantitatively.
+
+# 1 INTRODUCTION
+
+With the significant progress of recent machine learning research, various machine learning models have been being rapidly adopted to countless real-world applications. This rapid adaptation increasingly questions the machine learning model’s credibility, fairness, and more generally interpretability. In the line of this research, researchers have explored various notions of model interpretability. Some researchers directly answer the trustability (Ribeiro et al., 2016) or the fairness of a model (Zhao et al., 2017), while some other researchers seek to actually improve the model’s performance by understanding the model’s weak points (Koh & Liang, 2017). Even though the goal of such various model interpretability tasks varies, vast majority of them are built upon extracting relevant features for a prediction, so called feature-based explanation.
+
+Feature-based explanation is commonly based on measuring the fidelity of the explanation to the model, which is essentially how close the sum of attribution scores for a set of features approximates the function value difference before and after removing the set of features. Depending on their design, the fidelity-based attribution evaluation varies: completeness (Sundararajan et al., 2017), sensitivity-n (Ancona et al., 2018), infidelity (Yeh et al., 2019), and causal local explanation metric (Plumb et al., 2018). The idea of smallest sufficient region (SSR) and smallest destroying region (SDR) (Fong & Vedaldi, 2017; Dabkowski & Gal, 2017) is worth noting because it considers the ranking of the feature attribution scores, not the actual score itself. Intuitively, for a faithful attribution score, removing the most salient features would naturally lead to a large difference in prediction score. Therefore, SDR-based evaluations measure how much the function value changes when the most high-valued salient features are removed.
+
+Although the aforementioned attribution evaluations made success in many cases, setting features with an arbitrary reference values to zero-out the input is limited, in the sense that it only considers the prediction at the reference value while ignoring the rest of the input space. Furthermore, the choice of reference value inherently introduces bias. For example, if we set the feature value to 0 in rgb images, this introduces a bias in the attribution map that favors the bright pixels. As a result, explanations that optimize upon such evaluations often omit important dark objects and the pertinent negative features in the image, which is the part of the image that does not contain object but is crucial to the prediction (Dhurandhar et al., 2018). An alternative way to remove pixels is to use sampling from some predefined distribution or a generative model (Chang et al., 2018), which nevertheless could still introduce some bias with respect to the defined distribution. Moreover, they require a generative model that approximates the data distribution, which may not be available in certain domains.
+
+In this paper, we remove such inherit bias by taking a different perspective on the input perturbation. We start from an intuition that if a set of features are important to make a specific prediction, keeping them in the same values would preserve the prediction even though other irrelevant features are modified. In other words, the model would be more sensitive on the changes of those important or relevant features than the ones that are not. Unlike the foremost approaches including SDR and SSR that perturbs features to a specific reference point, we consider the minimum norm of perturbation to arbitrary directions, not just to a reference point, that can change model’s prediction, also known as “minimum adversarial perturbation” in the literature (Goodfellow et al., 2014; Weng et al., 2018b).
+
+Based on this idea, we define new evaluation criteria to test the importance of a set of features. By computing the minimum adversarial perturbation on the complementary set of features that can alter the model’s decision, we could test the degree of importance of the set. Although explicitly computing the importance value is NP-hard (Katz et al., 2017), Carlini & Wagner (2017) and Madry et al. (2017) showed that the perturbations computed by adversarial attacks can serve as reasonably tight upper bounds, which lead to an efficient approximation for the proposed evaluation.
+
+Furthermore, we can derive a new explanation framework by formulating the model explanation to a two-player min-max game between explanator and adversarial attacker. The explanator aims to find a set of important features to maximize the minimum perturbation computed by the attacker. This framework empirically performs much better than previous approaches quantitatively, with very inspiring examples.
+
+To summarize our contributions:
+
+• We define new evaluation criteria for feature-based explanations based on robustness analysis. The evaluation criteria consider the worst case perturbations when a set of features are anchored, which does not introduce bias into the evaluation.   
+We design efficient algorithms to generate explanations that maximize the proposed criteria, which perform favorably against baseline methods on the proposed evaluation criteria.   
+Experiments in computer vision and NLP models demonstrate that the proposed explanation can indeed identify some important features that are not captured by previous methods. Furthermore, our method is able to extract a set of features that contrasts the current prediction to a target class.
+
+# 2 ROBUSTNESS ANALYSIS FOR EVALUATING FEATURE-BASED EXPLANATIONS
+
+# 2.1 PROBLEM NOTATION
+
+Let us consider the following setting: a general $K$ -way classification problem with input space $\mathcal { X } \subseteq \mathbb { R } ^ { d }$ , output space $\mathcal { V } = \{ 1 , \ldots , K \}$ , and a predictor function $f : \mathcal { X }  \mathcal { Y }$ where $f ( { \pmb x } )$ denotes the output class for some input example $\pmb { x } = [ \bar { \pmb { x } } _ { 1 } , \dots , \pmb { x } _ { d } ] \in \mathcal { X }$ . Then, for a particular prediction $f ( { \pmb x } ) = { \boldsymbol y }$ , despite the different forms of existing feature-based explanations ranging from attributing an importance value to each feature, ranking the features by their importance, to simply identify a set of important features, a common goal of them is to extract a compact set of relevant features with respect to the prediction.
+
+# 2.2 EVALUATION THROUGH ROBUSTNESS ANALYSIS
+
+We note that however, given an explanation that identifies a set of said to be relevant features, how can we evaluate the quality of such explanation, or in other words, justify whether the distinguished features are truly relevant to the prediction? While one generally has no ground truth about the underlying true relevance of the features, recent studies take an axiomatic approach to define what properties the relevant features should hold and evaluate the explanations through verifying if the identified relevant features satisfy the properties. One such properties that is widely adopted in the literature is to assume that the importance of a set of features corresponds to the degree of change in prediction when the features are removed from the original input. Nevertheless, as we discussed in the previous section, the practice of approximating removal of features by setting their value to some reference point poses the risk of introducing bias in the evaluation. As a result, to escape from the caveat, we follow a similar concept but propose two new criteria to evaluate the importance of features based on the following assumptions.
+
+Assumption 1 When the values of the most salient features are anchored (fixed), perturbation on the complementary set of features has weaker influence on the model’s prediction. In other words, the model could tolerate a larger degree of perturbation on the less important and non-anchored features.
+
+Assumption 2 If perturbation is allowed on a set of important features, a small perturbation could easily change the model prediction even when we fix the values for the rest of the features.
+
+Based on these two assumptions, we propose a new framework for evaluating explanations. The evaluation is based on the adversarial robustness when a set of features are fixed, which is formally defined below.
+
+Definition 2.1 The minimum adversarial perturbation norm on a set of features $S$ , which we will also name as Robustness- $S$ , can be defined as:
+
+$$
+\epsilon _ { S } ^ { * } = g ( \pmb { x } , S ) = \{ \operatorname* { m i n } _ { \pmb { \delta } } \| \pmb { \delta } \| _ { p } s . t . \ f ( \pmb { x } + \pmb { \delta } ) \neq y , \ \pmb { \delta } _ { \overline { { S } } } = 0 \} ,
+$$
+
+where ${ \overline { { S } } } = U \setminus S$ is the complementary set of features, and $\delta _ { \overline { { S } } } = 0$ means that the perturbation value on features in $\overline { S }$ is constraint to be $O$ .
+
+Assume that we are given an explanation that partitions the input features into a relevant set $S _ { r }$ and an irrelevant set $\overline { { S _ { r } } }$ . Assumption 1 implies that the quality of the relevant set can be measured by $\epsilon _ { S _ { r } } ^ { * } -$ the robustness of irrelevant that a higher robustness on when the relevant set is anchored. Specifically, Assumptiofollows from a larger coverage of pertinent features in set $\overline { { S _ { r } } }$ $S _ { r }$ and thus an explanation is considered better if it leads to a higher robustness against perturbation in $\overline { { S _ { r } } }$ . On the other hand, based on Assumption 2, an explanation that has included important salient features in $S _ { r }$ should lead to a smaller robustness level on $\epsilon _ { S _ { r } } ^ { * }$ . Therefore, Assumption 1 and 2 build up our proposed evaluation criteria Robustness- $\overline { { S _ { r } } }$ and Robustness- $S _ { r }$ respectively, as listed below.
+
+Robustness- $\overline { { S _ { r } } }$ measures the minimum adversarial distortion $\epsilon _ { \overline { { S _ { r } } } }$ when the set of important features $S _ { r }$ , typically represented by the high-weight features in an attribution map, are anchored and perturbation is only allowed in low-weight regions. The higher the score the better the explanation. To measure Robustness- $\overline { { S _ { r } } }$ , we would need to first determine the size of $\lvert S _ { r } \rvert$ . We can set $\lvert S _ { r } \rvert$ to the amount of anchors that an user is interested in or we may vary the size of $\lvert S _ { r } \rvert$ and evaluate the corresponding Robustness- $\overline { { \boldsymbol { S } _ { r } } }$ at different points. By varying the size of $\lvert S _ { r } \rvert$ , we could plot an evaluation curve for each explanation and in turn measure the area under curve (AUC), which corresponds to the average Robustness- $\overline { { S _ { r } } }$ at different sizes of relevant set.
+
+Robustness- $S _ { r }$ measures the minimum distortion distance $\epsilon _ { S _ { r } }$ when the set of important features $S _ { r }$ are the only region that is perturbable, and the rest of feature values are anchored. Contrary to Robustness- $\overrightarrow { S _ { r } }$ , lower scores on this metric indicate better explanation. We similarly define AUC of Robustness- $S _ { r }$ as the average of Robustness- $\overline { { S _ { r } } }$ when we vary the size of $\lvert S _ { r } \rvert$ .
+
+Evaluation dinality of rocedure Note that both Robustness-. For instance, including all features $\overline { { S _ { r } } }$ nd Robustnwill make $S _ { r }$ e sensitive to the car-. Therefore, we will $S _ { r }$ $S _ { r }$ $\epsilon _ { S _ { r } } ^ { * } = 0$ a feature attribution method that assigns a weight with each feature, we can sort the features by the decending order of weights and then for each set of top- $K$ features with $K = 1 , 2 , \ldots , d$ , we evaluate Robustness- $\overline { { S _ { r } } } ( S _ { r } )$ and plot a curve. A larger (smaller) area under curve indicates a better feature attribution ranking. (See examples in Figure 1).
+
+Untargeted v.s. Targeted Explanation Definition 2.1 corresponds to the untargeted adversarial robustness – a perturbation that changes the predicted class to any label except $y$ is considered as a successful attack. Instead of doing this, our formulation can also extend to targeted adversarial robustness, where we replace (1) by
+
+$$
+\epsilon _ { S , t } ^ { * } = \lbrace \operatorname* { m i n } _ { \delta } \| \delta \| _ { p } \mathrm { ~ s . t . ~ } f ( \pmb { x } + \delta ) = t ; \delta _ { \overline { { S } } } = 0 \rbrace ,
+$$
+
+where $t$ is the targeted class. Using this definition, our approach will try to address the question “Why is this example classified as $y$ instead of $t ^ { \ast }$ , and the important features that optimize this criterion will highlight the contrast between class $y$ and $t$ . We will give several interesting results in the experiment section.
+
+Comparing to existing measurement The proposed criteria at the first glance look similar to SSR- and SDR-based measurements. We note that, however, the key differences between our proposed criteria and SSR- (SDR-) based criteria are in two-folds: 1) Conceptually, to measure whether a set of features is important, instead of concerning the prediction change before and after removing the features, we consider whether perturbation on the feature values would significantly alter the prediction. 2) Practically, our proposed criteria allow us to eschew the difficulty of modeling feature removal as discussed in section 1. In fact, as most implementations of removal-based criteria set the values of the features of interest to some fixed reference point, our criteria could be viewed as generalized versions where we consider all possible reference points by allowing perturbations in any directions. As a result, the proposed criteria enjoys a broader view of prediction behavior around the input, and in turn could capture a broader range of important features like the pertinent negative features in Dhurandhar et al. (2018), as we shall show in the experiment section.
+
+Robustness Evaluation under Fixed Anchor Set It is known that computing the exact minimum distortion distance in modern neural networks is intractable (Katz et al., 2017), so many different methods have been developed to estimate the value. Adversarial attacks, such as C&W (Carlini & Wagner, 2017) and PGD attack (Madry et al., 2017), aim to find a feasible solution of (1), which leads to an upper bound of $\epsilon _ { S } ^ { * }$ . They are based on gradient based optimizers which are usually efficient. On the other hand, neural network verification methods aim to provide a lower bound of $\epsilon _ { S } ^ { * }$ to ensure that the model prediction will not change within certain perturbation range (Singh et al., 2018; Wong & Kolter, 2018; Weng et al., 2018a; Gehr et al., 2018; Zhang et al., 2018; Wang et al., 2018; Zhang et al., 2019). However, these methods are usually time consuming (often $> 5 0$ times slower than a backpropagation).
+
+The proposed framework can be combined with any method that aims to approximately compute (1), including attack, verification, and some other statistical estimations. However, for simplicity we only choose to evaluate (1) by the state-of-the-art projected gradient descent (PGD) attack (Madry et al., 2017), since the verification methods are too slow and often lead to much looser estimation as reported in some recent studies (Salman et al., 2019).
+
+# 3 NEW EXPLANATIONS TOWARDS OPTIMIZING THE CRITERIA
+
+Given the new evaluation criteria, a natural follow-up question is how to design explanations that optimize the measurements. Recall that under the proposed criteria the goal of an optimal explanation is to maximize (minimize) robustness- $\bar { S } _ { r }$ (robustness- $S _ { r }$ ) under the cardinality constraint on $S _ { r }$ . Searching for such explanations thus leads to the following optimization problems, (3) for Robustness- $\overline { { S _ { r } } }$ and (4) for Robustness- $S _ { r }$ :
+
+$$
+\begin{array} { r l } { \underset { S _ { r } \in \{ 0 , 1 \} ^ { d } } { \mathrm { m a x i m i z e } } } & { { } g ( \pmb { x } , \pmb { S _ { r } } ) } \\ { \mathrm { s u b j e c t t o } } & { { } \lVert \pmb { S _ { r } } \rVert _ { 0 } \leq K , } \end{array}
+$$
+
+$$
+\begin{array} { r l } { \underset { S _ { r } \in \{ 0 , 1 \} ^ { d } } { \mathrm { m i n i m i z e } } } & { { } g ( \pmb { x } , S _ { r } ) } \\ { \mathrm { s u b j e c t ~ t o } } & { { } \| S _ { r } \| _ { 0 } \leq K , } \end{array}
+$$
+
+where $g ( { \pmb x } , S )$ computes the value in Eq. (1), the minimum distortion distance when the features in set $\bar { S _ { r } }$ is not allowed to be perturbed, and $K$ is a pre-defined size constraint on the set $S _ { r }$ .
+
+# 3.1 GREEDY ALGORITHM
+
+Directly solving (3) and (4) is challenging since $g$ is an implicit function computed by solving (1) approximately, and furthermore, the discrete input constraint makes it intractible to find the optimal solution. As a result, we propose a greedy-styled algorithm, where we iteratively add the most promising feature into $S _ { r }$ that optimizes the objective at each local step until $S _ { r }$ reaches the size constraint. In other words, we initialize the set $S _ { r }$ as empty, and sequentially solve the following subproblem at every step $t$ :
+
+$$
+\arg \operatorname* { m a x } _ { i } \ g ( { \pmb x } , \overline { { S _ { r } ^ { t } \cup i } } ) , \ \mathrm { o r \ a r g m i n } \ g ( { \pmb x } , S _ { r } ^ { t } \cup i ) , \ \forall i \in \overline { { S _ { r } } }
+$$
+
+where $S _ { r } ^ { t }$ is the anchor set at step $t$ , and $S _ { r } ^ { 0 } \ = \ \varnothing$ . We repeat this subprocedure until the size of set $S _ { r } ^ { t }$ reaches $K$ . We name this method as Greedy. A straightforward way for solving (5) is to exhaustively search over every single feature. However, considering a single feature at a time ignores the correlation between features, which tends to introduce noise (see our experimental results). If we consider multiple features at a single step, searching over all possible combinations will become intractable. For example, considering all possible combinations of two features requires $O ( d ^ { 2 } )$ evaluations of function $g$ at every step. To consider the joint influence between features efficiently, we propose a smoothed regression version of solving (5) in the following subsection.
+
+Table 1: Area under curve of the proposed criteria for various explanations on MNIST. The higher the better for Robustness- $\overline { { S _ { r } } }$ ; the lower the better for Robustness- $S _ { r }$ . Robustness measured with (1).   
+
+<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LO0</td><td>BBMP</td><td>Reg-Greedy</td><td>Greedy</td><td>One-Step Reg</td></tr><tr><td>Robustness-Sr</td><td>88.00</td><td>85.98</td><td>75.48</td><td>76.59</td><td>81.31</td><td>98.01</td><td>83.57</td><td>86.37</td></tr><tr><td>Robustness-Sr</td><td>91.72</td><td>91.97</td><td>101.49</td><td>98.82</td><td>173.90</td><td>82.81</td><td>171.56</td><td>83.59</td></tr></table>
+
+# 3.2 REGRESSION GREEDY
+
+As considering the correlation between features by searching over all possible subsets of $\overline { { S _ { r } ^ { t } } }$ at every step $t$ is computationally infeasible, we instead propose to approximate the function $g$ by learning a mapping from the binary space of $\{ 0 , 1 \} ^ { d }$ , where ones indicate the inclusion of corresponding feature indices and zeros otherwise, to their resulting function value $g ( x , \{ 0 , 1 \} ^ { d } )$ . Specifically, we can sample a subset $Q \subseteq \{ 0 , 1 \} ^ { d }$ and then consider the following linear regression:
+
+$$
+\begin{array} { r } { \pmb { w } ^ { * } = \underset { \pmb { w } } { \arg \operatorname* { m i n } } \sum _ { z \in Q } ( \pmb { w } ^ { T } z - g ( \pmb { x } , z ) ) ^ { 2 } . } \end{array}
+$$
+
+After the regression is learned, we can treat the coefficients $w$ that correspond to each feature as their approximated effect on the function value of $g$ when they are included into the set $S _ { r }$ . By learning such regression where we sample from the possible subsets, we are able to capture the joint relationships between features, and as well smooth out possible noises. In fact, the greedy approach can be viewed as a special case of Reg-Greedy where the sampled subset $Q$ , in (6), in each iterative step contains exactly the one-hot encoded vectors with the “on” indices correspond to the remaining feature indices. That is, each one-hot vector indicates the inclusion of a corresponding single feature into the relevant set. In this case, the coefficients of the learned linear regression would be equivalent to the difference in objective value before and after the corresponding feature is included into the relevant set. To take into account feature interactions, Reg-Greedy samples from the whole distribution of $\{ 0 , 1 \} ^ { d }$ where most of the sampled vectors in $Q$ contains multiple “on” indices. In this way, the learned regression captures feature correlations on the objective value and could smooth out possible noises encountered by greedy. There has been a great line of research on studying the interaction between features including the well-known Shapley value which tackles the problem through cooperative game theory perspective. And Lundberg $\&$ Lee (2017) proposed a way to use regression with a special kernel to approximate the Shapley value. However, sampling from the whole distribution of $\{ 0 , 1 \} ^ { d }$ could still incur exponential complexity, and using only a reasonable amount of samples might not be able to precisely capture the behavior of the highly nonlinear objective function $g$ . Therefore, we propose the Regression Greedy (Reg-Greedy) approach, where we still run greedy steps to incrementally add indices to $S _ { r } ^ { t }$ , but at each iteration we run this regression and use the weights to decide which index to be added to $S _ { r } ^ { t }$ . Note that at each step the samples $Q$ must be in a restricted domain, where indices that are already chosen in $S _ { r } ^ { t }$ should be 1 and we sample 0/1 only for the rest of the indices. We distinguish Reg-Greedy from onestep regression (One-Step Reg) which directly determines the importance of each feature by merely solving (6) once. By combining regression in a greedy procedure, we are able to gradually narrow down our sampling space (by sampling only from a restricted domain), focusing on the feature interactions between remaining features and the ones that are already added into the relevant set. This enables us to find from the remaining features that have the greatest interaction with the current relevant set, and could in turn maximally optimize the objective value when added into the relevant set. In practice, a sample complexity of $O ( d )$ for learning the regression could generally work well.
+
+# 4 EXPERIMENTS
+
+We present both qualitative and quantitative comparisons in the experiments. For $\| \cdot \| _ { p }$ in (1) and (2), we consider $p = 2$ , i.e., the $\ell _ { 2 }$ norm for all experiments. In quantitative results, including evaluation curves and the corresponding AUC, we report the average over 50 random examples. For
+
+![](images/78ddf98df023e2f9064b8764376c57f67ec1d5861f0f3b846c5e1912f0f43236.jpg)  
+Figure 1: Different Robustness- $\overline { { \boldsymbol { S } _ { r } } }$ (left) with varying $| \overline { { S _ { r } } } |$ and Robustness- $S _ { r }$ (right) with varying $\lvert S _ { r } \rvert$ . For Robustness- $\overline { { \boldsymbol { S } _ { r } } }$ (left), the higher the better; for Robustness- $S _ { r }$ (right), the lower the better. We omit points in the plot with value too high to fit in the scale of y-axis.
+
+![](images/319dfe339927d9000e4f953b4db6a64b0981eab7c2c40617f4959df017d643ed.jpg)  
+Figure 2: Visualization on our proposed methods. The top features selected by RegGreedy are less noisy.
+
+Table 2: Area under curve of the proposed criteria for various explanations on ImageNet. The higher the better for Robustness- $\overline { { S _ { r } } }$ ; the lower the better for Robustness- $S _ { r }$ . Robustness measured with (1).   
+
+<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LO0</td><td>BBMP</td><td>Reg-Greedy</td><td>Greedy</td><td>One-Step Reg</td></tr><tr><td>Robustness-Sr</td><td>27.13</td><td>26.01</td><td>18.25</td><td>23.54</td><td>22.60</td><td>31.62</td><td>21.16</td><td>24.54</td></tr><tr><td>Robustness-Sr</td><td>45.53</td><td>46.28</td><td>60.02</td><td>52.77</td><td>154.14</td><td>43.97</td><td>58.45</td><td>47.07</td></tr></table>
+
+Table 3: Area under curve of the Insertion and Deletion criteria for various explanations on MNIST. The higher the better for Insertion; the lower the better for Deletion.   
+
+<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LOO</td><td>BBMP</td><td>Reg-Greedy</td></tr><tr><td>Insertion</td><td>250.81</td><td>262.74</td><td>200.50</td><td>192.44</td><td>102.53</td><td>379.15</td></tr><tr><td>Deletion</td><td>281.88</td><td>273.71</td><td>362.68</td><td>442.65</td><td>527.80</td><td>159.77</td></tr></table>
+
+<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LO0</td><td>BBMP</td><td>Reg-Greedy</td></tr><tr><td>Rank Correlation</td><td>0.3001</td><td>0.3042</td><td>0.1108</td><td>0.4966</td><td>0.1775</td><td>0.1835</td></tr></table>
+
+Table 4: Rank correlation between explanations with respect to original and randomized model.
+
+the proposed algorithms, we consider Reg-Greedy (Sec 3.2), One-Step Reg (Sec 3.2) and Greedy (Sec 3.1). For other baselines we include vanilla gradient (Grad) (Shrikumar et al., 2017) and integrated gradient (IG) (Sundararajan et al., 2017) from gradient-based approaches; leave-one-out (LOO), or occlusion-1, (Zeiler & Fergus, 2014; Li et al., 2016) and SHAP (Lundberg & Lee, 2017) from perturbation-based approaches (Ancona et al., 2018), and black-box meaningful perturbation (BBMP) (Fong & Vedaldi, 2017) from SSR/SDR-based approaches. We perform our experiments on two image datasets, MNIST and ImageNet, as well as a text dataset YahooAnswers.
+
+# 4.1 QUANTITATIVE ANALYSIS ACROSS DIFFERENT EXPLANATIONS
+
+The proposed measurements: Robustness- $\overline { { S _ { r } } }$ and Robustness- $S _ { r }$ . We compare different explanations under the two proposed criteria, robustness- $\overline { { \boldsymbol { S } _ { r } } }$ and robustness- $S _ { r }$ , and plot their evaluation curves respectively. For ease of comparison, we calculate the area under curve (AUC) for each corresponding evaluation. We list the results in Table 3, and leave the plots in appendix A. As shown in Table 3, under both criteria, comparing to regression-based methods, the pure greedy method usually suffers degraded performances that could be due to the ignorance of feature correlations, which ultimately results in the introduction of noise as shown in Figure 2. Furthermore from the table, we observe that the proposed regression-greedy method consistently outperforms others on both criteria. On one hand, this suggests that the proposed algorithm indeed successfully optimizes towards the criteria; on the other hand, this might indicate the proposed criteria do capture different characteristics of explanations which most of the current explanations do not possess. Another somewhat interesting finding from the table is that while vanilla gradient has generally been viewed as a baseline method, it nonetheless performs competitively on the proposed criteria. To investigate deeper into such observation, we shall visualize the explanations in the following subsection. For simplicity we will just apply Reg-Greedy with Robustness- $\overline { { S _ { r } } }$ criterion in the qualitative comparisons with previous methods.
+
+![](images/19dadb04565472600b2b783b24ffa9d0b6ce9d6e45e53e10cfd0da99a67fba6a.jpg)  
+Figure 3: Visualization on top 20 percent relevant features provided by existing explanations.
+
+![](images/d8485dd939bb049d7c55d1e02da8154749142dad7484fd7e843a689b578920da.jpg)
+
+![](images/c62a8ec5ad06303bad9449fe739478410224dab4bfc59e300907876e6c931d9c.jpg)  
+Figure 4: Visualization of targeted explanation. In each row, we highlight relevant regions explaining why the input is not predicted as the target class.   
+Figure 5: Comparisons between different targeted explanations against different targeted class on MNIST.
+
+Existing commonly adopted measurements: Insertion and Deletion. Indeed, it might not be surprising that Reg-Greedy achieves the best performances on the proposed criteria it is explicitly designed to optimize. To more objectively showcase the usefulness of the proposed explanation, we compare Reg-Greedy with other explanations on existing commonly used quantitative measurements. Particularly, we adopt the Deletion and Insertion criteria proposed by Petsiuk et al. (2018), which are generalized variants of the region perturbation criterion presented in Samek et al. (2016). The Deletion criterion measures the probability drop in the predicted class as top-relevant features, indicated by the given explanation, are progressively removed from the input. On the other hand, the Insertion criterion measures the increase in probability of the predicted class as top-relevant features are gradually revealed from the input whose features are originally all masked. Similar to our proposed criteria, a quick drop (and thus a small area under curve) or a sharp increase (that leads to a large area under curve) in Deletion and Insertion respectively suggest a good explanation as the selected top-important features could indeed greatly influence the prediction. In the experiments, we follow Samek et al. (2016) to remove features by setting their values to randomly sampled values. We plot the evaluation curves and report corresponding AUCs in Figure 12 and Table 3. On these additional two criteria, we observe that our proposed method consistently performs favorably against other explanations.
+
+Sanity Check. As pointed out in recent literature that an appropriate explanation should at least be loosely related the model being explained (Adebayo et al., 2018), to ensure that our proposed explanation does indeed reflect the model behavior, we conduct the sanity check proposed by (Adebayo et al., NeurIPS’18) to check if our explanations are adequately different when the model parameters are randomly re-initialized. In the experiment, we randomly re-initialize the last fully-connected layer of the neural network model. We then compute the rank correlation between explanation computed w.r.t. the original model and that w.r.t. the randomized model. From Table 4, we observe that Reg-Greedy has a much lower rank correlation comparing to Grad, IG, and LOO, suggesting that Reg-Greedy is indeed sensitive to model parameter change and is able to pass the sanity check.
+
+# 4.2 QUALITATIVE VISUALIZATIONS
+
+Visualized Explanations on MNIST. Figure 3 illustrates the top features identified by various explanation methods. From this figure, we observe that Gradient, IG, SHAP mainly highlights the white pixels in the digit, while gradient and IG are more noisy compared to SHAP. In the contrary, Reg-Greedy focuses on both the “crucial positive” of the digits “pertinent negative” of regions around the digit. For example, in the first row, a 7 might have been predicted as a 4 or 0 if the pixels highlighted by Reg-Greedy are set to 1. Similarly, a 1 may be turned to a 4 or a 7 given additional white pixels to its left, and a 9 may become a 7 if deleted the lower circular part of its head. As a result, Reg-Greedy focuses on “the region in which perturbing will lead to easier prediction change”, which includes both the crucial positive pixels and pertinent negative pixels, and provides additional insights that are not captured by the baseline explanations. The superiority of Reg-Greedy is also validated by the better performance on the Robustness- $\overline { { S _ { r } } }$ score.
+
+Targeted Explanation. Recall that in section 2.2, we discussed about the possibility of defining the robustness measurement by considering a targeted distortion distance as formulated in (2). Here, we provide examples, as shown in Figure 4, where we answer the question of “why the input digit is an A but not a $B ^ { \prime \prime }$ by defining a targeted perturbation distance towards class B as our robustness measurement. In each row of the figure, we provide targeted explanation towards two different target classes for a same input image. Interestingly, as the target classes changes, the generated explanation varies in an interpretatble way. For example, in the first row, we explain why the input digit 7 is not classified as a 9 (middle column) or a 2 (rightmost column). The resulting explanation against 9 highlights the upper-left part of the 7. Semantically, this region is indeed pertinent to the classification between 7 and 9, since turning on the highlighted pixel values in the region (currently black in the original image) will then make the 7 resemble a 9. However, the targeted explanation against 2 highlights a very different but also meaningful region, which is the lower-right part of the 7; since adding a horizontal stroke on the area would turn a 7 into a 2. This finding demonstrates a special characteristic of our explanation which cannot be easily found in most of the existing methods.
+
+While the capability of capturing not only the crucial positive but also the pertinent negative features have also been observed in some recently proposed explanations such as Layer-wise Relevance Propagation (LRP) (Bach et al., 2015), as reported in Samek et al. (2016), as well as the explanation technique proposed in Oramas et al. (2019). Both of the above mentioned methods are not explicitly designed to handle the targeted explanation task which attempt to answer the question “what are the important features that lead to the prediction of class A but not class $\mathbf { B } ^ { \ast }$ , and thus has different limitations. For example, the ability of LRP to capture pertinent negative features in fact heavily depends on the input range. In Samek et al. (2016) where inputs are normalized to have zero mean and a standard deviation of one, the black background will have non-zero value, and LRP would have non-zero attributions on the black background pixels which allows the explanation to capture pertinent negative features. However, as later shown in Dhurandhar et al. (2018), if the input pixel intensity is normalized into the range between 0 and 1 (where background pixels have the values of 0), LRP failed to highlight the pertinent negative pixels, as background would always have zero attribution (since LRP is equivalent to multiplication between Grad and input in a Rectified Linear Unit (ReLU) network as shown in Ancona et al. (2018)). In Oramas et al. (2019), unlike our targeted explanation where we know exactly which targeted class the explanation is suggesting against (and by varying the targeted class we observe varying corresponding explanation given), their method by design does not convey such information. The pertinent negative features highlighted by their method by construction is not directly related to a specific target class, and users in fact need to infer what target class the pertinent negative features are preventing against. To further grasp the difference, we compare our explanation with theirs in Figure 5 (we borrow the results from Oramas et al. (2019) for visualization of their method). Qualitatively, we also observe that our method seems to be giving the most natural explanations. For example, in the first row of left image where the highlighted features are against the class 0, in addition to the left vertical gap (which when presence would make 2 looks like a 0) that is roughly highlighted by all three methods, our method is the only one that highlights the right tail part (green circled) of the digit 2 which might also serve as crucial evidence of 2 against 0. Furthermore, as we change the targeted class to 7 (the second row), while LRP seems to be providing similar explanations, we observe that our explanation has a drastic change and highlights the green circled part which when turned off will make 2 becomes a 7. These results might suggest our method is more capable of handling such targeted explanation task.
+
+![](images/e1725234b1202f08c3d0527bcf5c0856e14ebcfd82c94e03bab763982eaf2a72.jpg)  
+Figure 6: Visualization of different explanations on ImageNet, where the predicted class for each input is “fish”, “bird”, “dog”, and “sea lion”.
+
+![](images/e4ae02892a79e3f7573287e21a3a7af51fdc28390ba75d1450bdba25b3dd101a.jpg)  
+Figure 7: Explanations on a text classification model where the predicted label for this sentence is “sport”.
+
+Visualized Explanations on ImageNet. On ImageNet, we as well compare different explanations quantitatively on both of the proposed criteria. We plot the evaluation curves (in appendix A), and compute the corresponding AUC, as listed in Table 2. In general, we observe similar trends as the experiments shown in MNIST. In particular, Reg-Greedy enjoys an overall superior performances than existing explanations on the criteria. In addition, several visualization results in Figure 6 also qualitatively demonstrate that our method provides more compact explanations that focuses more on the actual object being classified.
+
+Text Classification. We demonstrate how our explanation method could be applied to text classification models. Note that a length- $\mathbf { \nabla } \cdot n$ sentence is usually represented by $n$ embedding vectors, and thus when applying our Greedy algorithm, at each iteration we will try to add each embedding vector to the set $S _ { r }$ and choose the one with largest reward. Since there are only at most $n$ choices, the Greedy algorithm doesn’t suffer much from noise and has similar behavior with Reg-Greedy.
+
+We perform experiments on an LSTM network which learns to classify a given sentence into one of the ten classes (Society, Science, Health, . . . ). We showcase an example with explanations generated with different methods in Figure 7. We note that although the top-5 relevant keyword sets generated by the three methods do not vary much, the rankings within the highlighted keywords for each explanation are in fact different. We observe that our method Greedy tends to generate explanation that matches human intuition the most. Particularly, to predict the label of “sport”, one might consider “cleats”, “football”, and “cut” as the strongest indications towards the concept “sport”.
+
+# 5 RELATED WORK
+
+Our work proposes an objective measurement of feature-based explanation by measuring the “minimum adversarial perturbation” in adversarial literature, which is estimated by adversarial attack. We provide a necessarily incomplete review on related works in objective measurement of explanations and adversarial robustness.
+
+Objective Measurements for Explanations Evaluation of explanations has been a difficult problem mainly due to the absence of ground truth (Ancona et al., 2018; Sundararajan et al., 2017). Although one could rely on human intuitions to assess the quality of the generated explanations (Lundberg & Lee, 2017; Doshi-Velez & Kim, 2017), for example, judging whether the explanation focuses on the object of interest in an image classification task, these evaluations subject to human perceptions are prone to fall into the pitfall of favoring user-friendly explanations, such as attributions that visually aligns better with the input image, which might not reflect the model behavior (Adebayo et al., 2018). As a result, in addition to subjective measurements, recent literature has also proposed objective measurements, which is also called functionally-grounded evaluations (DoshiVelez & Kim, 2017). We roughly categorize existing objective measurements into two families.
+
+This first family of explanation evaluation is called fidelity-based measurement. This includes that Completeness or Sum to Delta which requires the sum of attributions to equal the prediction difference of the original input and baseline (Sundararajan et al., 2017; Shrikumar et al., 2017); sensitivity-n which further generalizes completeness to any subset of the feature (Ancona et al., 2018); local accuracy Ribeiro et al. (2016); Lundberg & Lee (2017); and infidelity which is a framework that encompasses several (Yeh et al., 2019). The general philosophy for this line of methods is to require the sum of attribution value faithfully reflect the change in prediction function value given the presence or absence of certain subset of features. The second family of explanation evaluation are removal-based and preservation-based measurements, which focus on identifying the most important set of features with respect to a particular prediction. The underlying assumption made is that by removing the most (least) salient feature, the resulting function value should drop (increase) the most. (Samek et al., 2016) proposed this idea as an evaluation to evaluate the ranking of featureattribution score. Later on, Fong & Vedaldi (2017) derive explanations by solving an optimization problem to optimize the evaluation. And Dabkowski & Gal (2017) proposed to learn the explanation generating process by training an auxiliary model.
+
+We note the implicitly in the evaluation process of both fidelity and removal (preservation) based measurement involves computing the change in function value given some set of features being absent. However, it is difficult to carefully model the concept of feature absence in practice, as most models by construction are not able to handle inputs with real missing features. As a result, previous work has compromised by using approximation to estimate the effect of removing certain features. This includes setting the values of the features to be removed by zero (Ancona et al., 2018; Sundararajan et al., 2017) or the mean value (Lundberg & Lee, 2017), blurred value (Fong & Vedaldi, 2017), random value (Samek et al., 2016; Dabkowski & Gal, 2017), or more advanced generative model that attempts to model the given data distribution (Chang et al., 2018). Unfortunately, such approximations that represent feature absence by setting the their values to some predefined distribution would inevitably introduce bias into the evaluation process. With the presence of this inherent caveat, we are thus inspired to adopt another angle to tackle the explanation problem.
+
+Adversarial Robustness Adversarial robustness has been extensively studied in the past few years. The adversarial robustness of a machine learning model on a given sample can be defined as the shortest distance from the sample to the decision boundary, which corresponds to our definition in (1). Algorithms have been proposed for finding adversarial examples (feasible solutions of (1)), including (Goodfellow et al., 2014; Carlini & Wagner, 2017; Madry et al., 2017). However, those algorithms only work for neural networks, while for other models such as tree based models or nearest neighbor classifiers, adversarial examples can be found by decision based attacks (Brendel et al., 2017; Cheng et al., 2018; Chen et al., 2019). Therefore the proposed framework can also be used in other decision based classifiers. On the other hand, several works aim to solve the neural network verification problem, which is equivalent to finding a lower bound of (1). Examples include (Singh et al., 2018; Wong & Kolter, 2018; Zhang et al., 2018). In principal, our work can also apply these verification methods for getting an approximate solution of (1), but in practice they are very slow to run and often gives loose lower bounds on regular trained networks.
+
+Our work is also closely related to related works that consider the question ”For situation A, why was the outcome B and not $\mathbf { { C } } ^ { \ast }$ , which we call counterfactual explanations. Xu et al. (2018) add group sparsity regularization to adversarial attack to enforce semantic structure for the perturbation, which is more interpretable. Ribeiro et al. (2018) find a set of features that once fixed, probability of the prediction is high when perturbing other features. Goyal et al. (2019) show how one could change the input feature such that the system would output a different class, where the change is limited to replacing a part of input feature by a part of an distractor image. Dhurandhar et al. (2018) consider the pertinent negative in a binary setting by solving a carefully designed loss function.
+
+# 6 CONCLUSION
+
+In this paper, we establish the link between a set of features to a prediction with a new evaluation criteria, robustness analysis, which measures the minimum tolerance of adversarial perturbation. Furthermore, we develop a new explanation method to find important set of features to optimize this new criterion. Experimental results demonstrate that the proposed new explanations are indeed capturing significant feature sets across multiple domains.
+
+# REFERENCES
+
+Julius Adebayo, Justin Gilmer, Michael Muelly, Ian Goodfellow, Moritz Hardt, and Been Kim. Sanity checks for saliency maps. In Advances in Neural Information Processing Systems, pp. 9525–9536, 2018.
+
+Marco Ancona, Enea Ceolini, Cengiz Oztireli, and Markus Gross. A unified view of gradient-based attribution methods for deep neural networks. International Conference on Learning Representations, 2018.
+
+Sebastian Bach, Alexander Binder, Gre´goire Montavon, Frederick Klauschen, Klaus-Robert Mu¨ller, and Wojciech Samek. On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation. PloS one, 10(7):e0130140, 2015.
+
+Wieland Brendel, Jonas Rauber, and Matthias Bethge. Decision-based adversarial attacks: Reliable attacks against black-box machine learning models. arXiv preprint arXiv:1712.04248, 2017.
+
+Nicholas Carlini and David Wagner. Towards evaluating the robustness of neural networks. In 2017 IEEE Symposium on Security and Privacy $( S P )$ , pp. 39–57. IEEE, 2017.
+
+Chun-Hao Chang, Elliot Creager, Andrew A. Goldenberg, and David Kristjanson Duvenaud. Explaining image classifiers by counterfactual generation. In ICLR 2019, 2018.
+
+Hongge Chen, Huan Zhang, Duane Boning, and Cho-Jui Hsieh. Robust decision trees against adversarial examples. In ICML, 2019.
+
+Minhao Cheng, Thong Le, Pin-Yu Chen, Jinfeng Yi, Huan Zhang, and Cho-Jui Hsieh. Queryefficient hard-label black-box attack: An optimization-based approach. arXiv preprint arXiv:1807.04457, 2018.
+
+Piotr Dabkowski and Yarin Gal. Real time image saliency for black box classifiers. In NIPS, 2017.
+
+Amit Dhurandhar, Pin-Yu Chen, Ronny Luss, Chun-Chen Tu, Paishun Ting, Karthikeyan Shanmugam, and Payel Das. Explanations based on the missing: Towards contrastive explanations with pertinent negatives. In Advances in Neural Information Processing Systems, pp. 592–603, 2018.
+
+Finale Doshi-Velez and Been Kim. Towards a rigorous science of interpretable machine learning. arXiv preprint arXiv:1702.08608, 2017.
+
+Ruth C. Fong and Andrea Vedaldi. Interpretable explanations of black boxes by meaningful perturbation. 2017 IEEE International Conference on Computer Vision (ICCV), pp. 3449–3457, 2017.
+
+Timon Gehr, Matthew Mirman, Dana Drachsler-Cohen, Petar Tsankov, Swarat Chaudhuri, and Martin Vechev. Ai2: Safety and robustness certification of neural networks with abstract interpretation. In 2018 IEEE Symposium on Security and Privacy $( S P )$ , pp. 3–18. IEEE, 2018.
+
+Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. arXiv preprint arXiv:1412.6572, 2014.
+
+Yash Goyal, Ziyan Wu, Jan Ernst, Dhruv Batra, Devi Parikh, and Stefan Lee. Counterfactual visual explanations. In International Conference on Machine Learning, pp. 2376–2384, 2019.
+
+Guy Katz, Clark Barrett, David L Dill, Kyle Julian, and Mykel J Kochenderfer. Reluplex: An efficient smt solver for verifying deep neural networks. In International Conference on Computer Aided Verification, pp. 97–117. Springer, 2017.
+
+Pang Wei Koh and Percy Liang. Understanding black-box predictions via influence functions. In International Conference on Machine Learning, pp. 1885–1894, 2017.
+
+Jiwei Li, Will Monroe, and Dan Jurafsky. Understanding neural networks through representation erasure. CoRR, abs/1612.08220, 2016. URL http://arxiv.org/abs/1612.08220.
+
+Scott M Lundberg and Su-In Lee. A unified approach to interpreting model predictions. In Advances in Neural Information Processing Systems, pp. 4765–4774, 2017.
+
+Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. arXiv preprint arXiv:1706.06083, 2017.
+
+Jose Oramas, Kaili Wang, and Tinne Tuytelaars. Visual explanation by interpretation: Improving visual feedback capabilities of deep neural networks. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ H1ziPjC5Fm.
+
+Vitali Petsiuk, Abir Das, and Kate Saenko. Rise: Randomized input sampling for explanation of black-box models. arXiv preprint arXiv:1806.07421, 2018.
+
+Gregory Plumb, Denali Molitor, and Ameet S Talwalkar. Model agnostic supervised local explanations. In Advances in Neural Information Processing Systems, pp. 2515–2524, 2018.
+
+Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Why should i trust you?: Explaining the predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1135–1144. ACM, 2016.
+
+Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Anchors: High-precision model-agnostic explanations. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.
+
+Hadi Salman, Greg Yang, Huan Zhang, Cho-Jui Hsieh, and Pengchuan Zhang. A convex relaxation barrier to tight robust verification of neural networks. arXiv preprint arXiv:1902.08722, 2019.
+
+Wojciech Samek, Alexander Binder, Gre´goire Montavon, Sebastian Lapuschkin, and Klaus-Robert Mu¨ller. Evaluating the visualization of what a deep neural network has learned. IEEE transactions on neural networks and learning systems, 28(11):2660–2673, 2016.
+
+Avanti Shrikumar, Peyton Greenside, and Anshul Kundaje. Learning important features through propagating activation differences. International Conference on Machine Learning, 2017.
+
+Gagandeep Singh, Timon Gehr, Matthew Mirman, Markus Pu¨schel, and Martin Vechev. Fast and effective robustness certification. In Advances in Neural Information Processing Systems, pp. 10802–10813, 2018.
+
+Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. In International Conference on Machine Learning, 2017.
+
+Shiqi Wang, Kexin Pei, Justin Whitehouse, Junfeng Yang, and Suman Jana. Efficient formal safety analysis of neural networks. In Advances in Neural Information Processing Systems, pp. 6367– 6377, 2018.
+
+Tsui-Wei Weng, Huan Zhang, Hongge Chen, Zhao Song, Cho-Jui Hsieh, Luca Daniel, Duane Boning, and Inderjit Dhillon. Towards fast computation of certified robustness for relu networks. In International Conference on Machine Learning, pp. 5273–5282, 2018a.
+
+Tsui-Wei Weng, Huan Zhang, Pin-Yu Chen, Jinfeng Yi, Dong Su, Yupeng Gao, Cho-Jui Hsieh, and Luca Daniel. Evaluating the robustness of neural networks: An extreme value theory approach. arXiv preprint arXiv:1801.10578, 2018b.
+
+Eric Wong and Zico Kolter. Provable defenses against adversarial examples via the convex outer adversarial polytope. In International Conference on Machine Learning, pp. 5283–5292, 2018.
+
+Kaidi Xu, Sijia Liu, Pu Zhao, Pin-Yu Chen, Huan Zhang, Quanfu Fan, Deniz Erdogmus, Yanzhi Wang, and Xue Lin. Structured adversarial attack: Towards general implementation and better interpretability. arXiv preprint arXiv:1808.01664, 2018.
+
+Chih-Kuan Yeh, Cheng-Yu Hsieh, Arun Sai Suggala, David I. Inouye, and Pradeep Ravikumar. On the (in)fidelity and sensitivity for explanations. CoRR, abs/1901.09392, 2019.
+
+Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pp. 818–833. Springer, 2014.
+
+Huan Zhang, Tsui-Wei Weng, Pin-Yu Chen, Cho-Jui Hsieh, and Luca Daniel. Efficient neural network robustness certification with general activation functions. In Advances in neural information processing systems, pp. 4939–4948, 2018.
+
+Huan Zhang, Pengchuan Zhang, and Cho-Jui Hsieh. Recurjac: An efficient recursive algorithm for bounding jacobian matrix of neural networks and its applications. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 5757–5764, 2019.
+
+Jieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, and Kai-Wei Chang. Men also like shopping: Reducing gender bias amplification using corpus-level constraints. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pp. 2979–2989, 2017.
+
+# A EVALUATION CURVES
+
+![](images/a47860d5bb1b45e69b2d70e32d53f2d9373bbd412d6d920222f16b83980ea971.jpg)  
+Figure 8: Comparisons between our proposed methods under different criteria. From left to right: untargeted Robustness- $\overline { { \boldsymbol { S } _ { r } } }$ , targeted Robustness- $\overline { { \boldsymbol { S } _ { r } } }$ , untargeted Robustness- $S _ { r }$ , targeted Robustness$S _ { r }$ . We omit points in the plot with value too high to fit in the scale of y-axis.
+
+![](images/5727086bf61d07de10d03f21470c816f1a5373474f0ae8d8ee1a8a41cf50af51.jpg)  
+Figure 9: Comparisons between our proposed methods and existing explanations under different criteria. From left to right: untargeted Robustness- $\overline { { \boldsymbol { S } _ { r } } }$ , targeted Robustness- $\overline { { S _ { r } } }$ , untargeted Robustness$S _ { r }$ , targeted Robustness- $S _ { r }$ . We omit points in the plot with value too high to fit in the scale of y-axis.
+
+![](images/a9350862af1406cdbf707a1ca40e6693eccfed8ac8ff4da0f9bb839c321ebe2c.jpg)  
+Figure 10: Comparisons between our proposed methods under different criteria on ImageNet. From left to right: untargeted Robustness- $\overline { { \boldsymbol { S } _ { r } } }$ , targeted Robustness- $\overline { { \boldsymbol { S } _ { r } } }$ , untargeted Robustness- $S _ { r }$ , targeted Robustness- $S _ { r }$ . We omit points in the plot with value too high to fit in the scale of y-axis.
+
+![](images/5666adba4deefe3b02d28f9206b3f84130a620acfba171f8c8c6a60f558f8b96.jpg)  
+Figure 11: Comparisons between our proposed methods under different criteria on ImageNet. From left to right: untargeted Robustness- $\vec { \cdot S _ { r } }$ , targeted Robustness- $\overline { { \boldsymbol { S } _ { r } } }$ , untargeted Robustness- $S _ { r }$ , targeted Robustness- $S _ { r }$ . We omit points in the plot with value too high to fit in the scale of y-axis.
+
+![](images/a925714ae0ea3b8327a809c6052b8e0494eefd2f6cc6eedef6f86b6406c0636a.jpg)  
+Figure 12: Comparisons between explanations under different criteria on MNIST. Left figure: change in output logits as relevant features are inserted into the input. Right figure: change in output logits as relevant features are removed from the input.
+
+<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LOO</td><td>BBMP</td><td>Reg-Greedy</td></tr><tr><td>Insertion</td><td>250.81</td><td>262.74</td><td>200.50</td><td>192.44</td><td>102.53</td><td>379.15</td></tr><tr><td>Deletion</td><td>281.88</td><td>273.71</td><td>362.68</td><td>442.65</td><td>527.80</td><td>159.77</td></tr></table>
+
+Table 5: Area under curve of the Insertion and Deletion criteria for various explanations on MNIST.   
+The higher the better for Insertion; the lower the better for Deletion.
+
+# C T-TEST ON AUC
+
+Table 6: The proposed Reg-Greedy versus other explanations on MNIST under our proposed criteria with Student’s $t$ -test at $9 5 \%$ confidence level.   
+
+<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LOO</td><td>BBMP</td></tr><tr><td>Robustness-Sr</td><td>win</td><td>win</td><td>win</td><td>win</td><td>win</td></tr><tr><td>Robustness-Sr</td><td>win</td><td>win</td><td>win</td><td>win</td><td>win</td></tr></table>
+
+Table 7: The proposed Reg-Greedy versus other explanations on MNIST under Insertion and Deletion criteria with Student’s $t$ -test at $9 5 \%$ confidence level.   
+
+<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LO0</td><td>BBMP</td></tr><tr><td>Insertion</td><td>win</td><td>win</td><td>win</td><td>win</td><td>win</td></tr><tr><td>Deletion</td><td>win</td><td>win</td><td>win</td><td>win</td><td>win</td></tr></table>
+
+# D SANITY CHECK
+
+<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LOO</td><td>BBMP</td><td>Reg-Greedy</td></tr><tr><td>Rank Correlation</td><td>0.3001</td><td>0.3042</td><td>0.1108</td><td>0.4966</td><td>0.1775</td><td>0.1835</td></tr></table>
+
+Table 8: Rank correlation between explanations with respect to original and randomized model.
+
+# E COMPARISONS ON TARGETED EXPLANATION
+
+![](images/051a251c11e28f15ca92221d21bf76f0f346a9a48a57ecbfefc10c81a8ea8634.jpg)  
+Figure 13: Comparisons between different targeted explanations against different targeted class on MNIST.
+
+# F HEATMAP VISUALIZATION ON MNIST
+
+![](images/85b4204194e7049b8f7a2b9e33fb6a6fc348a91de3e493ae5d47e24b19e67520.jpg)  
+Figure 14: Heatmap Visualization of different explanations on MNIST.

parse/train/Hye4KeSYDr/Hye4KeSYDr_content_list.json

@@ -0,0 +1,1718 @@
+[
+    {
+        "type": "text",
+        "text": "EVALUATIONS AND METHODS FOR EXPLANATION THROUGH ROBUSTNESS ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            772,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Among multiple ways of interpreting a machine learning model, measuring the importance of a set of features tied to a prediction is probably one of the most intuitive way to explain a model. In this paper, we establish the link between a set of features to a prediction with a new evaluation criterion, robustness analysis, which measures the minimum distortion distance of adversarial perturbation. By measuring the tolerance level for an adversarial attack, we can extract a set of features that provides the most robust support for a current prediction, and also can extract a set of features that contrasts the current prediction to a target class by setting a targeted adversarial attack. By applying this methodology to various prediction tasks across multiple domains, we observe the derived explanations are indeed capturing the significant feature set qualitatively and quantitatively. ",
+        "bbox": [
+            233,
+            265,
+            764,
+            417
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            441,
+            336,
+            457
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "With the significant progress of recent machine learning research, various machine learning models have been being rapidly adopted to countless real-world applications. This rapid adaptation increasingly questions the machine learning model’s credibility, fairness, and more generally interpretability. In the line of this research, researchers have explored various notions of model interpretability. Some researchers directly answer the trustability (Ribeiro et al., 2016) or the fairness of a model (Zhao et al., 2017), while some other researchers seek to actually improve the model’s performance by understanding the model’s weak points (Koh & Liang, 2017). Even though the goal of such various model interpretability tasks varies, vast majority of them are built upon extracting relevant features for a prediction, so called feature-based explanation. ",
+        "bbox": [
+            174,
+            465,
+            825,
+            590
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Feature-based explanation is commonly based on measuring the fidelity of the explanation to the model, which is essentially how close the sum of attribution scores for a set of features approximates the function value difference before and after removing the set of features. Depending on their design, the fidelity-based attribution evaluation varies: completeness (Sundararajan et al., 2017), sensitivity-n (Ancona et al., 2018), infidelity (Yeh et al., 2019), and causal local explanation metric (Plumb et al., 2018). The idea of smallest sufficient region (SSR) and smallest destroying region (SDR) (Fong & Vedaldi, 2017; Dabkowski & Gal, 2017) is worth noting because it considers the ranking of the feature attribution scores, not the actual score itself. Intuitively, for a faithful attribution score, removing the most salient features would naturally lead to a large difference in prediction score. Therefore, SDR-based evaluations measure how much the function value changes when the most high-valued salient features are removed. ",
+        "bbox": [
+            174,
+            597,
+            825,
+            750
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although the aforementioned attribution evaluations made success in many cases, setting features with an arbitrary reference values to zero-out the input is limited, in the sense that it only considers the prediction at the reference value while ignoring the rest of the input space. Furthermore, the choice of reference value inherently introduces bias. For example, if we set the feature value to 0 in rgb images, this introduces a bias in the attribution map that favors the bright pixels. As a result, explanations that optimize upon such evaluations often omit important dark objects and the pertinent negative features in the image, which is the part of the image that does not contain object but is crucial to the prediction (Dhurandhar et al., 2018). An alternative way to remove pixels is to use sampling from some predefined distribution or a generative model (Chang et al., 2018), which nevertheless could still introduce some bias with respect to the defined distribution. Moreover, they require a generative model that approximates the data distribution, which may not be available in certain domains. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we remove such inherit bias by taking a different perspective on the input perturbation. We start from an intuition that if a set of features are important to make a specific prediction, keeping them in the same values would preserve the prediction even though other irrelevant features are modified. In other words, the model would be more sensitive on the changes of those important or relevant features than the ones that are not. Unlike the foremost approaches including SDR and SSR that perturbs features to a specific reference point, we consider the minimum norm of perturbation to arbitrary directions, not just to a reference point, that can change model’s prediction, also known as “minimum adversarial perturbation” in the literature (Goodfellow et al., 2014; Weng et al., 2018b). ",
+        "bbox": [
+            174,
+            103,
+            825,
+            214
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Based on this idea, we define new evaluation criteria to test the importance of a set of features. By computing the minimum adversarial perturbation on the complementary set of features that can alter the model’s decision, we could test the degree of importance of the set. Although explicitly computing the importance value is NP-hard (Katz et al., 2017), Carlini & Wagner (2017) and Madry et al. (2017) showed that the perturbations computed by adversarial attacks can serve as reasonably tight upper bounds, which lead to an efficient approximation for the proposed evaluation. ",
+        "bbox": [
+            174,
+            222,
+            823,
+            305
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we can derive a new explanation framework by formulating the model explanation to a two-player min-max game between explanator and adversarial attacker. The explanator aims to find a set of important features to maximize the minimum perturbation computed by the attacker. This framework empirically performs much better than previous approaches quantitatively, with very inspiring examples. ",
+        "bbox": [
+            174,
+            313,
+            825,
+            382
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To summarize our contributions: ",
+        "bbox": [
+            176,
+            388,
+            387,
+            404
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We define new evaluation criteria for feature-based explanations based on robustness analysis. The evaluation criteria consider the worst case perturbations when a set of features are anchored, which does not introduce bias into the evaluation.   \nWe design efficient algorithms to generate explanations that maximize the proposed criteria, which perform favorably against baseline methods on the proposed evaluation criteria.   \nExperiments in computer vision and NLP models demonstrate that the proposed explanation can indeed identify some important features that are not captured by previous methods. Furthermore, our method is able to extract a set of features that contrasts the current prediction to a target class. ",
+        "bbox": [
+            215,
+            415,
+            825,
+            549
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 ROBUSTNESS ANALYSIS FOR EVALUATING FEATURE-BASED EXPLANATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            568,
+            705,
+            599
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 PROBLEM NOTATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            609,
+            361,
+            623
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Let us consider the following setting: a general $K$ -way classification problem with input space $\\mathcal { X } \\subseteq \\mathbb { R } ^ { d }$ , output space $\\mathcal { V } = \\{ 1 , \\ldots , K \\}$ , and a predictor function $f : \\mathcal { X }  \\mathcal { Y }$ where $f ( { \\pmb x } )$ denotes the output class for some input example $\\pmb { x } = [ \\bar { \\pmb { x } } _ { 1 } , \\dots , \\pmb { x } _ { d } ] \\in \\mathcal { X }$ . Then, for a particular prediction $f ( { \\pmb x } ) = { \\boldsymbol y }$ , despite the different forms of existing feature-based explanations ranging from attributing an importance value to each feature, ranking the features by their importance, to simply identify a set of important features, a common goal of them is to extract a compact set of relevant features with respect to the prediction. ",
+        "bbox": [
+            174,
+            627,
+            825,
+            726
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 EVALUATION THROUGH ROBUSTNESS ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            743,
+            547,
+            757
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We note that however, given an explanation that identifies a set of said to be relevant features, how can we evaluate the quality of such explanation, or in other words, justify whether the distinguished features are truly relevant to the prediction? While one generally has no ground truth about the underlying true relevance of the features, recent studies take an axiomatic approach to define what properties the relevant features should hold and evaluate the explanations through verifying if the identified relevant features satisfy the properties. One such properties that is widely adopted in the literature is to assume that the importance of a set of features corresponds to the degree of change in prediction when the features are removed from the original input. Nevertheless, as we discussed in the previous section, the practice of approximating removal of features by setting their value to some reference point poses the risk of introducing bias in the evaluation. As a result, to escape from the caveat, we follow a similar concept but propose two new criteria to evaluate the importance of features based on the following assumptions. ",
+        "bbox": [
+            174,
+            761,
+            825,
+            928
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1 When the values of the most salient features are anchored (fixed), perturbation on the complementary set of features has weaker influence on the model’s prediction. In other words, the model could tolerate a larger degree of perturbation on the less important and non-anchored features. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            160
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2 If perturbation is allowed on a set of important features, a small perturbation could easily change the model prediction even when we fix the values for the rest of the features. ",
+        "bbox": [
+            171,
+            166,
+            823,
+            195
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Based on these two assumptions, we propose a new framework for evaluating explanations. The evaluation is based on the adversarial robustness when a set of features are fixed, which is formally defined below. ",
+        "bbox": [
+            176,
+            202,
+            821,
+            244
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1 The minimum adversarial perturbation norm on a set of features $S$ , which we will also name as Robustness- $S$ , can be defined as: ",
+        "bbox": [
+            173,
+            255,
+            823,
+            284
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e00aab4efdf6a1570127b3028accb3993e12b346535b7292af75e3d255cabf91.jpg",
+        "text": "$$\n\\epsilon _ { S } ^ { * } = g ( \\pmb { x } , S ) = \\{ \\operatorname* { m i n } _ { \\pmb { \\delta } } \\| \\pmb { \\delta } \\| _ { p } s . t . \\ f ( \\pmb { x } + \\pmb { \\delta } ) \\neq y , \\ \\pmb { \\delta } _ { \\overline { { S } } } = 0 \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            287,
+            686,
+            310
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\overline { { S } } } = U \\setminus S$ is the complementary set of features, and $\\delta _ { \\overline { { S } } } = 0$ means that the perturbation value on features in $\\overline { S }$ is constraint to be $O$ . ",
+        "bbox": [
+            174,
+            316,
+            823,
+            348
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assume that we are given an explanation that partitions the input features into a relevant set $S _ { r }$ and an irrelevant set $\\overline { { S _ { r } } }$ . Assumption 1 implies that the quality of the relevant set can be measured by $\\epsilon _ { S _ { r } } ^ { * } -$ the robustness of irrelevant that a higher robustness on when the relevant set is anchored. Specifically, Assumptiofollows from a larger coverage of pertinent features in set $\\overline { { S _ { r } } }$ $S _ { r }$ and thus an explanation is considered better if it leads to a higher robustness against perturbation in $\\overline { { S _ { r } } }$ . On the other hand, based on Assumption 2, an explanation that has included important salient features in $S _ { r }$ should lead to a smaller robustness level on $\\epsilon _ { S _ { r } } ^ { * }$ . Therefore, Assumption 1 and 2 build up our proposed evaluation criteria Robustness- $\\overline { { S _ { r } } }$ and Robustness- $S _ { r }$ respectively, as listed below. ",
+        "bbox": [
+            173,
+            358,
+            825,
+            479
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Robustness- $\\overline { { S _ { r } } }$ measures the minimum adversarial distortion $\\epsilon _ { \\overline { { S _ { r } } } }$ when the set of important features $S _ { r }$ , typically represented by the high-weight features in an attribution map, are anchored and perturbation is only allowed in low-weight regions. The higher the score the better the explanation. To measure Robustness- $\\overline { { S _ { r } } }$ , we would need to first determine the size of $\\lvert S _ { r } \\rvert$ . We can set $\\lvert S _ { r } \\rvert$ to the amount of anchors that an user is interested in or we may vary the size of $\\lvert S _ { r } \\rvert$ and evaluate the corresponding Robustness- $\\overline { { \\boldsymbol { S } _ { r } } }$ at different points. By varying the size of $\\lvert S _ { r } \\rvert$ , we could plot an evaluation curve for each explanation and in turn measure the area under curve (AUC), which corresponds to the average Robustness- $\\overline { { S _ { r } } }$ at different sizes of relevant set. ",
+        "bbox": [
+            173,
+            486,
+            825,
+            603
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Robustness- $S _ { r }$ measures the minimum distortion distance $\\epsilon _ { S _ { r } }$ when the set of important features $S _ { r }$ are the only region that is perturbable, and the rest of feature values are anchored. Contrary to Robustness- $\\overrightarrow { S _ { r } }$ , lower scores on this metric indicate better explanation. We similarly define AUC of Robustness- $S _ { r }$ as the average of Robustness- $\\overline { { S _ { r } } }$ when we vary the size of $\\lvert S _ { r } \\rvert$ . ",
+        "bbox": [
+            174,
+            611,
+            825,
+            669
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Evaluation dinality of rocedure Note that both Robustness-. For instance, including all features $\\overline { { S _ { r } } }$ nd Robustnwill make $S _ { r }$ e sensitive to the car-. Therefore, we will $S _ { r }$ $S _ { r }$ $\\epsilon _ { S _ { r } } ^ { * } = 0$ a feature attribution method that assigns a weight with each feature, we can sort the features by the decending order of weights and then for each set of top- $K$ features with $K = 1 , 2 , \\ldots , d$ , we evaluate Robustness- $\\overline { { S _ { r } } } ( S _ { r } )$ and plot a curve. A larger (smaller) area under curve indicates a better feature attribution ranking. (See examples in Figure 1). ",
+        "bbox": [
+            173,
+            676,
+            825,
+            779
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Untargeted v.s. Targeted Explanation Definition 2.1 corresponds to the untargeted adversarial robustness – a perturbation that changes the predicted class to any label except $y$ is considered as a successful attack. Instead of doing this, our formulation can also extend to targeted adversarial robustness, where we replace (1) by ",
+        "bbox": [
+            173,
+            785,
+            825,
+            842
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1cab98b051184bc7eb49327b4e892e01b63848f1ee47ad31795d3f34ce141da0.jpg",
+        "text": "$$\n\\epsilon _ { S , t } ^ { * } = \\lbrace \\operatorname* { m i n } _ { \\delta } \\| \\delta \\| _ { p } \\mathrm { ~ s . t . ~ } f ( \\pmb { x } + \\delta ) = t ; \\delta _ { \\overline { { S } } } = 0 \\rbrace ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            845,
+            651,
+            867
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $t$ is the targeted class. Using this definition, our approach will try to address the question “Why is this example classified as $y$ instead of $t ^ { \\ast }$ , and the important features that optimize this criterion will highlight the contrast between class $y$ and $t$ . We will give several interesting results in the experiment section. ",
+        "bbox": [
+            174,
+            872,
+            823,
+            929
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Comparing to existing measurement The proposed criteria at the first glance look similar to SSR- and SDR-based measurements. We note that, however, the key differences between our proposed criteria and SSR- (SDR-) based criteria are in two-folds: 1) Conceptually, to measure whether a set of features is important, instead of concerning the prediction change before and after removing the features, we consider whether perturbation on the feature values would significantly alter the prediction. 2) Practically, our proposed criteria allow us to eschew the difficulty of modeling feature removal as discussed in section 1. In fact, as most implementations of removal-based criteria set the values of the features of interest to some fixed reference point, our criteria could be viewed as generalized versions where we consider all possible reference points by allowing perturbations in any directions. As a result, the proposed criteria enjoys a broader view of prediction behavior around the input, and in turn could capture a broader range of important features like the pertinent negative features in Dhurandhar et al. (2018), as we shall show in the experiment section. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            270
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Robustness Evaluation under Fixed Anchor Set It is known that computing the exact minimum distortion distance in modern neural networks is intractable (Katz et al., 2017), so many different methods have been developed to estimate the value. Adversarial attacks, such as C&W (Carlini & Wagner, 2017) and PGD attack (Madry et al., 2017), aim to find a feasible solution of (1), which leads to an upper bound of $\\epsilon _ { S } ^ { * }$ . They are based on gradient based optimizers which are usually efficient. On the other hand, neural network verification methods aim to provide a lower bound of $\\epsilon _ { S } ^ { * }$ to ensure that the model prediction will not change within certain perturbation range (Singh et al., 2018; Wong & Kolter, 2018; Weng et al., 2018a; Gehr et al., 2018; Zhang et al., 2018; Wang et al., 2018; Zhang et al., 2019). However, these methods are usually time consuming (often $> 5 0$ times slower than a backpropagation). ",
+        "bbox": [
+            174,
+            286,
+            825,
+            426
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The proposed framework can be combined with any method that aims to approximately compute (1), including attack, verification, and some other statistical estimations. However, for simplicity we only choose to evaluate (1) by the state-of-the-art projected gradient descent (PGD) attack (Madry et al., 2017), since the verification methods are too slow and often lead to much looser estimation as reported in some recent studies (Salman et al., 2019). ",
+        "bbox": [
+            174,
+            431,
+            825,
+            502
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 NEW EXPLANATIONS TOWARDS OPTIMIZING THE CRITERIA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            517,
+            694,
+            532
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given the new evaluation criteria, a natural follow-up question is how to design explanations that optimize the measurements. Recall that under the proposed criteria the goal of an optimal explanation is to maximize (minimize) robustness- $\\bar { S } _ { r }$ (robustness- $S _ { r }$ ) under the cardinality constraint on $S _ { r }$ . Searching for such explanations thus leads to the following optimization problems, (3) for Robustness- $\\overline { { S _ { r } } }$ and (4) for Robustness- $S _ { r }$ : ",
+        "bbox": [
+            174,
+            540,
+            825,
+            611
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f848ebb8cf49c087ee367a23837c1a7b9e211a7486cdbcdcc853f85ddf8e329.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { S _ { r } \\in \\{ 0 , 1 \\} ^ { d } } { \\mathrm { m a x i m i z e } } } & { { } g ( \\pmb { x } , \\pmb { S _ { r } } ) } \\\\ { \\mathrm { s u b j e c t t o } } & { { } \\lVert \\pmb { S _ { r } } \\rVert _ { 0 } \\leq K , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            616,
+            419,
+            664
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8b77f343f1078dead8c7b4f353f807bff8fb4599896a230b84b2be13b5e1cc43.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { S _ { r } \\in \\{ 0 , 1 \\} ^ { d } } { \\mathrm { m i n i m i z e } } } & { { } g ( \\pmb { x } , S _ { r } ) } \\\\ { \\mathrm { s u b j e c t ~ t o } } & { { } \\| S _ { r } \\| _ { 0 } \\leq K , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            576,
+            616,
+            743,
+            664
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $g ( { \\pmb x } , S )$ computes the value in Eq. (1), the minimum distortion distance when the features in set $\\bar { S _ { r } }$ is not allowed to be perturbed, and $K$ is a pre-defined size constraint on the set $S _ { r }$ . ",
+        "bbox": [
+            173,
+            669,
+            826,
+            698
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 GREEDY ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            715,
+            364,
+            729
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Directly solving (3) and (4) is challenging since $g$ is an implicit function computed by solving (1) approximately, and furthermore, the discrete input constraint makes it intractible to find the optimal solution. As a result, we propose a greedy-styled algorithm, where we iteratively add the most promising feature into $S _ { r }$ that optimizes the objective at each local step until $S _ { r }$ reaches the size constraint. In other words, we initialize the set $S _ { r }$ as empty, and sequentially solve the following subproblem at every step $t$ : ",
+        "bbox": [
+            173,
+            741,
+            825,
+            825
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/01d2831037892758cb41d32b03dacfd8513249b55252ad9b4df3b27079e37ad9.jpg",
+        "text": "$$\n\\arg \\operatorname* { m a x } _ { i } \\ g ( { \\pmb x } , \\overline { { S _ { r } ^ { t } \\cup i } } ) , \\ \\mathrm { o r \\ a r g m i n } \\ g ( { \\pmb x } , S _ { r } ^ { t } \\cup i ) , \\ \\forall i \\in \\overline { { S _ { r } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            832,
+            691,
+            859
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $S _ { r } ^ { t }$ is the anchor set at step $t$ , and $S _ { r } ^ { 0 } \\ = \\ \\varnothing$ . We repeat this subprocedure until the size of set $S _ { r } ^ { t }$ reaches $K$ . We name this method as Greedy. A straightforward way for solving (5) is to exhaustively search over every single feature. However, considering a single feature at a time ignores the correlation between features, which tends to introduce noise (see our experimental results). If we consider multiple features at a single step, searching over all possible combinations will become intractable. For example, considering all possible combinations of two features requires $O ( d ^ { 2 } )$ evaluations of function $g$ at every step. To consider the joint influence between features efficiently, we propose a smoothed regression version of solving (5) in the following subsection. ",
+        "bbox": [
+            174,
+            866,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/9c547847ae8e6bc71e388c37faeaf97c870caf24e13ee0f8b83169a8c77eaa42.jpg",
+        "table_caption": [
+            "Table 1: Area under curve of the proposed criteria for various explanations on MNIST. The higher the better for Robustness- $\\overline { { S _ { r } } }$ ; the lower the better for Robustness- $S _ { r }$ . Robustness measured with (1). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LO0</td><td>BBMP</td><td>Reg-Greedy</td><td>Greedy</td><td>One-Step Reg</td></tr><tr><td>Robustness-Sr</td><td>88.00</td><td>85.98</td><td>75.48</td><td>76.59</td><td>81.31</td><td>98.01</td><td>83.57</td><td>86.37</td></tr><tr><td>Robustness-Sr</td><td>91.72</td><td>91.97</td><td>101.49</td><td>98.82</td><td>173.90</td><td>82.81</td><td>171.56</td><td>83.59</td></tr></table>",
+        "bbox": [
+            184,
+            101,
+            812,
+            157
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            220,
+            823,
+            276
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 REGRESSION GREEDY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            286,
+            367,
+            301
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As considering the correlation between features by searching over all possible subsets of $\\overline { { S _ { r } ^ { t } } }$ at every step $t$ is computationally infeasible, we instead propose to approximate the function $g$ by learning a mapping from the binary space of $\\{ 0 , 1 \\} ^ { d }$ , where ones indicate the inclusion of corresponding feature indices and zeros otherwise, to their resulting function value $g ( x , \\{ 0 , 1 \\} ^ { d } )$ . Specifically, we can sample a subset $Q \\subseteq \\{ 0 , 1 \\} ^ { d }$ and then consider the following linear regression: ",
+        "bbox": [
+            174,
+            306,
+            825,
+            377
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7b01e52729a9688c40c61fb12183b913009fa1094fc0a971692b995686abb564.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { w } ^ { * } = \\underset { \\pmb { w } } { \\arg \\operatorname* { m i n } } \\sum _ { z \\in Q } ( \\pmb { w } ^ { T } z - g ( \\pmb { x } , z ) ) ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            383,
+            640,
+            412
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "After the regression is learned, we can treat the coefficients $w$ that correspond to each feature as their approximated effect on the function value of $g$ when they are included into the set $S _ { r }$ . By learning such regression where we sample from the possible subsets, we are able to capture the joint relationships between features, and as well smooth out possible noises. In fact, the greedy approach can be viewed as a special case of Reg-Greedy where the sampled subset $Q$ , in (6), in each iterative step contains exactly the one-hot encoded vectors with the “on” indices correspond to the remaining feature indices. That is, each one-hot vector indicates the inclusion of a corresponding single feature into the relevant set. In this case, the coefficients of the learned linear regression would be equivalent to the difference in objective value before and after the corresponding feature is included into the relevant set. To take into account feature interactions, Reg-Greedy samples from the whole distribution of $\\{ 0 , 1 \\} ^ { d }$ where most of the sampled vectors in $Q$ contains multiple “on” indices. In this way, the learned regression captures feature correlations on the objective value and could smooth out possible noises encountered by greedy. There has been a great line of research on studying the interaction between features including the well-known Shapley value which tackles the problem through cooperative game theory perspective. And Lundberg $\\&$ Lee (2017) proposed a way to use regression with a special kernel to approximate the Shapley value. However, sampling from the whole distribution of $\\{ 0 , 1 \\} ^ { d }$ could still incur exponential complexity, and using only a reasonable amount of samples might not be able to precisely capture the behavior of the highly nonlinear objective function $g$ . Therefore, we propose the Regression Greedy (Reg-Greedy) approach, where we still run greedy steps to incrementally add indices to $S _ { r } ^ { t }$ , but at each iteration we run this regression and use the weights to decide which index to be added to $S _ { r } ^ { t }$ . Note that at each step the samples $Q$ must be in a restricted domain, where indices that are already chosen in $S _ { r } ^ { t }$ should be 1 and we sample 0/1 only for the rest of the indices. We distinguish Reg-Greedy from onestep regression (One-Step Reg) which directly determines the importance of each feature by merely solving (6) once. By combining regression in a greedy procedure, we are able to gradually narrow down our sampling space (by sampling only from a restricted domain), focusing on the feature interactions between remaining features and the ones that are already added into the relevant set. This enables us to find from the remaining features that have the greatest interaction with the current relevant set, and could in turn maximally optimize the objective value when added into the relevant set. In practice, a sample complexity of $O ( d )$ for learning the regression could generally work well. ",
+        "bbox": [
+            174,
+            419,
+            825,
+            835
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            856,
+            326,
+            872
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We present both qualitative and quantitative comparisons in the experiments. For $\\| \\cdot \\| _ { p }$ in (1) and (2), we consider $p = 2$ , i.e., the $\\ell _ { 2 }$ norm for all experiments. In quantitative results, including evaluation curves and the corresponding AUC, we report the average over 50 random examples. For ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/78ddf98df023e2f9064b8764376c57f67ec1d5861f0f3b846c5e1912f0f43236.jpg",
+        "image_caption": [
+            "Figure 1: Different Robustness- $\\overline { { \\boldsymbol { S } _ { r } } }$ (left) with varying $| \\overline { { S _ { r } } } |$ and Robustness- $S _ { r }$ (right) with varying $\\lvert S _ { r } \\rvert$ . For Robustness- $\\overline { { \\boldsymbol { S } _ { r } } }$ (left), the higher the better; for Robustness- $S _ { r }$ (right), the lower the better. We omit points in the plot with value too high to fit in the scale of y-axis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            114,
+            576,
+            227
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/319dfe339927d9000e4f953b4db6a64b0981eab7c2c40617f4959df017d643ed.jpg",
+        "image_caption": [
+            "Figure 2: Visualization on our proposed methods. The top features selected by RegGreedy are less noisy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            650,
+            108,
+            807,
+            228
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/90dae4bfdfd0b2e7953c992df9a9d75f5f85a26ac579bd89523eacf2b99c93d0.jpg",
+        "table_caption": [
+            "Table 2: Area under curve of the proposed criteria for various explanations on ImageNet. The higher the better for Robustness- $\\overline { { S _ { r } } }$ ; the lower the better for Robustness- $S _ { r }$ . Robustness measured with (1). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LO0</td><td>BBMP</td><td>Reg-Greedy</td><td>Greedy</td><td>One-Step Reg</td></tr><tr><td>Robustness-Sr</td><td>27.13</td><td>26.01</td><td>18.25</td><td>23.54</td><td>22.60</td><td>31.62</td><td>21.16</td><td>24.54</td></tr><tr><td>Robustness-Sr</td><td>45.53</td><td>46.28</td><td>60.02</td><td>52.77</td><td>154.14</td><td>43.97</td><td>58.45</td><td>47.07</td></tr></table>",
+        "bbox": [
+            186,
+            323,
+            812,
+            378
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/2efcb142d72576f246a697c30a4bd8fa6b5d9ff4c76c4396629421da6fc72362.jpg",
+        "table_caption": [
+            "Table 3: Area under curve of the Insertion and Deletion criteria for various explanations on MNIST. The higher the better for Insertion; the lower the better for Deletion. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LOO</td><td>BBMP</td><td>Reg-Greedy</td></tr><tr><td>Insertion</td><td>250.81</td><td>262.74</td><td>200.50</td><td>192.44</td><td>102.53</td><td>379.15</td></tr><tr><td>Deletion</td><td>281.88</td><td>273.71</td><td>362.68</td><td>442.65</td><td>527.80</td><td>159.77</td></tr></table>",
+        "bbox": [
+            256,
+            430,
+            738,
+            484
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/103e960dc7d4beb6c48dd91e0710389b594644f99c09839c58c19bf24e8fbb62.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LO0</td><td>BBMP</td><td>Reg-Greedy</td></tr><tr><td>Rank Correlation</td><td>0.3001</td><td>0.3042</td><td>0.1108</td><td>0.4966</td><td>0.1775</td><td>0.1835</td></tr></table>",
+        "bbox": [
+            243,
+            536,
+            750,
+            579
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Rank correlation between explanations with respect to original and randomized model. ",
+        "bbox": [
+            186,
+            588,
+            808,
+            603
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "the proposed algorithms, we consider Reg-Greedy (Sec 3.2), One-Step Reg (Sec 3.2) and Greedy (Sec 3.1). For other baselines we include vanilla gradient (Grad) (Shrikumar et al., 2017) and integrated gradient (IG) (Sundararajan et al., 2017) from gradient-based approaches; leave-one-out (LOO), or occlusion-1, (Zeiler & Fergus, 2014; Li et al., 2016) and SHAP (Lundberg & Lee, 2017) from perturbation-based approaches (Ancona et al., 2018), and black-box meaningful perturbation (BBMP) (Fong & Vedaldi, 2017) from SSR/SDR-based approaches. We perform our experiments on two image datasets, MNIST and ImageNet, as well as a text dataset YahooAnswers. ",
+        "bbox": [
+            174,
+            628,
+            825,
+            727
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 QUANTITATIVE ANALYSIS ACROSS DIFFERENT EXPLANATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            744,
+            650,
+            758
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The proposed measurements: Robustness- $\\overline { { S _ { r } } }$ and Robustness- $S _ { r }$ . We compare different explanations under the two proposed criteria, robustness- $\\overline { { \\boldsymbol { S } _ { r } } }$ and robustness- $S _ { r }$ , and plot their evaluation curves respectively. For ease of comparison, we calculate the area under curve (AUC) for each corresponding evaluation. We list the results in Table 3, and leave the plots in appendix A. As shown in Table 3, under both criteria, comparing to regression-based methods, the pure greedy method usually suffers degraded performances that could be due to the ignorance of feature correlations, which ultimately results in the introduction of noise as shown in Figure 2. Furthermore from the table, we observe that the proposed regression-greedy method consistently outperforms others on both criteria. On one hand, this suggests that the proposed algorithm indeed successfully optimizes towards the criteria; on the other hand, this might indicate the proposed criteria do capture different characteristics of explanations which most of the current explanations do not possess. Another somewhat interesting finding from the table is that while vanilla gradient has generally been viewed as a baseline method, it nonetheless performs competitively on the proposed criteria. To investigate deeper into such observation, we shall visualize the explanations in the following subsection. For simplicity we will just apply Reg-Greedy with Robustness- $\\overline { { S _ { r } } }$ criterion in the qualitative comparisons with previous methods. ",
+        "bbox": [
+            173,
+            770,
+            825,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/19dadb04565472600b2b783b24ffa9d0b6ce9d6e45e53e10cfd0da99a67fba6a.jpg",
+        "image_caption": [
+            "Figure 3: Visualization on top 20 percent relevant features provided by existing explanations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            122,
+            498,
+            265
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/d8485dd939bb049d7c55d1e02da8154749142dad7484fd7e843a689b578920da.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            581,
+            102,
+            771,
+            270
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/c62a8ec5ad06303bad9449fe739478410224dab4bfc59e300907876e6c931d9c.jpg",
+        "image_caption": [
+            "Figure 4: Visualization of targeted explanation. In each row, we highlight relevant regions explaining why the input is not predicted as the target class. ",
+            "Figure 5: Comparisons between different targeted explanations against different targeted class on MNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            337,
+            790,
+            452
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            515,
+            825,
+            585
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Existing commonly adopted measurements: Insertion and Deletion. Indeed, it might not be surprising that Reg-Greedy achieves the best performances on the proposed criteria it is explicitly designed to optimize. To more objectively showcase the usefulness of the proposed explanation, we compare Reg-Greedy with other explanations on existing commonly used quantitative measurements. Particularly, we adopt the Deletion and Insertion criteria proposed by Petsiuk et al. (2018), which are generalized variants of the region perturbation criterion presented in Samek et al. (2016). The Deletion criterion measures the probability drop in the predicted class as top-relevant features, indicated by the given explanation, are progressively removed from the input. On the other hand, the Insertion criterion measures the increase in probability of the predicted class as top-relevant features are gradually revealed from the input whose features are originally all masked. Similar to our proposed criteria, a quick drop (and thus a small area under curve) or a sharp increase (that leads to a large area under curve) in Deletion and Insertion respectively suggest a good explanation as the selected top-important features could indeed greatly influence the prediction. In the experiments, we follow Samek et al. (2016) to remove features by setting their values to randomly sampled values. We plot the evaluation curves and report corresponding AUCs in Figure 12 and Table 3. On these additional two criteria, we observe that our proposed method consistently performs favorably against other explanations. ",
+        "bbox": [
+            174,
+            602,
+            825,
+            837
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Sanity Check. As pointed out in recent literature that an appropriate explanation should at least be loosely related the model being explained (Adebayo et al., 2018), to ensure that our proposed explanation does indeed reflect the model behavior, we conduct the sanity check proposed by (Adebayo et al., NeurIPS’18) to check if our explanations are adequately different when the model parameters are randomly re-initialized. In the experiment, we randomly re-initialize the last fully-connected layer of the neural network model. We then compute the rank correlation between explanation computed w.r.t. the original model and that w.r.t. the randomized model. From Table 4, we observe that Reg-Greedy has a much lower rank correlation comparing to Grad, IG, and LOO, suggesting that Reg-Greedy is indeed sensitive to model parameter change and is able to pass the sanity check. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            160
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 QUALITATIVE VISUALIZATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            176,
+            433,
+            190
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Visualized Explanations on MNIST. Figure 3 illustrates the top features identified by various explanation methods. From this figure, we observe that Gradient, IG, SHAP mainly highlights the white pixels in the digit, while gradient and IG are more noisy compared to SHAP. In the contrary, Reg-Greedy focuses on both the “crucial positive” of the digits “pertinent negative” of regions around the digit. For example, in the first row, a 7 might have been predicted as a 4 or 0 if the pixels highlighted by Reg-Greedy are set to 1. Similarly, a 1 may be turned to a 4 or a 7 given additional white pixels to its left, and a 9 may become a 7 if deleted the lower circular part of its head. As a result, Reg-Greedy focuses on “the region in which perturbing will lead to easier prediction change”, which includes both the crucial positive pixels and pertinent negative pixels, and provides additional insights that are not captured by the baseline explanations. The superiority of Reg-Greedy is also validated by the better performance on the Robustness- $\\overline { { S _ { r } } }$ score. ",
+        "bbox": [
+            174,
+            202,
+            825,
+            354
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Targeted Explanation. Recall that in section 2.2, we discussed about the possibility of defining the robustness measurement by considering a targeted distortion distance as formulated in (2). Here, we provide examples, as shown in Figure 4, where we answer the question of “why the input digit is an A but not a $B ^ { \\prime \\prime }$ by defining a targeted perturbation distance towards class B as our robustness measurement. In each row of the figure, we provide targeted explanation towards two different target classes for a same input image. Interestingly, as the target classes changes, the generated explanation varies in an interpretatble way. For example, in the first row, we explain why the input digit 7 is not classified as a 9 (middle column) or a 2 (rightmost column). The resulting explanation against 9 highlights the upper-left part of the 7. Semantically, this region is indeed pertinent to the classification between 7 and 9, since turning on the highlighted pixel values in the region (currently black in the original image) will then make the 7 resemble a 9. However, the targeted explanation against 2 highlights a very different but also meaningful region, which is the lower-right part of the 7; since adding a horizontal stroke on the area would turn a 7 into a 2. This finding demonstrates a special characteristic of our explanation which cannot be easily found in most of the existing methods. ",
+        "bbox": [
+            174,
+            362,
+            825,
+            569
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "While the capability of capturing not only the crucial positive but also the pertinent negative features have also been observed in some recently proposed explanations such as Layer-wise Relevance Propagation (LRP) (Bach et al., 2015), as reported in Samek et al. (2016), as well as the explanation technique proposed in Oramas et al. (2019). Both of the above mentioned methods are not explicitly designed to handle the targeted explanation task which attempt to answer the question “what are the important features that lead to the prediction of class A but not class $\\mathbf { B } ^ { \\ast }$ , and thus has different limitations. For example, the ability of LRP to capture pertinent negative features in fact heavily depends on the input range. In Samek et al. (2016) where inputs are normalized to have zero mean and a standard deviation of one, the black background will have non-zero value, and LRP would have non-zero attributions on the black background pixels which allows the explanation to capture pertinent negative features. However, as later shown in Dhurandhar et al. (2018), if the input pixel intensity is normalized into the range between 0 and 1 (where background pixels have the values of 0), LRP failed to highlight the pertinent negative pixels, as background would always have zero attribution (since LRP is equivalent to multiplication between Grad and input in a Rectified Linear Unit (ReLU) network as shown in Ancona et al. (2018)). In Oramas et al. (2019), unlike our targeted explanation where we know exactly which targeted class the explanation is suggesting against (and by varying the targeted class we observe varying corresponding explanation given), their method by design does not convey such information. The pertinent negative features highlighted by their method by construction is not directly related to a specific target class, and users in fact need to infer what target class the pertinent negative features are preventing against. To further grasp the difference, we compare our explanation with theirs in Figure 5 (we borrow the results from Oramas et al. (2019) for visualization of their method). Qualitatively, we also observe that our method seems to be giving the most natural explanations. For example, in the first row of left image where the highlighted features are against the class 0, in addition to the left vertical gap (which when presence would make 2 looks like a 0) that is roughly highlighted by all three methods, our method is the only one that highlights the right tail part (green circled) of the digit 2 which might also serve as crucial evidence of 2 against 0. Furthermore, as we change the targeted class to 7 (the second row), while LRP seems to be providing similar explanations, we observe that our explanation has a drastic change and highlights the green circled part which when turned off will make 2 becomes a 7. These results might suggest our method is more capable of handling such targeted explanation task. ",
+        "bbox": [
+            174,
+            578,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e1725234b1202f08c3d0527bcf5c0856e14ebcfd82c94e03bab763982eaf2a72.jpg",
+        "image_caption": [
+            "Figure 6: Visualization of different explanations on ImageNet, where the predicted class for each input is “fish”, “bird”, “dog”, and “sea lion”. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            101,
+            483,
+            224
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/e4ae02892a79e3f7573287e21a3a7af51fdc28390ba75d1450bdba25b3dd101a.jpg",
+        "image_caption": [
+            "Figure 7: Explanations on a text classification model where the predicted label for this sentence is “sport”. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            522,
+            111,
+            805,
+            218
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            310,
+            823,
+            381
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Visualized Explanations on ImageNet. On ImageNet, we as well compare different explanations quantitatively on both of the proposed criteria. We plot the evaluation curves (in appendix A), and compute the corresponding AUC, as listed in Table 2. In general, we observe similar trends as the experiments shown in MNIST. In particular, Reg-Greedy enjoys an overall superior performances than existing explanations on the criteria. In addition, several visualization results in Figure 6 also qualitatively demonstrate that our method provides more compact explanations that focuses more on the actual object being classified. ",
+        "bbox": [
+            174,
+            388,
+            825,
+            486
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Text Classification. We demonstrate how our explanation method could be applied to text classification models. Note that a length- $\\mathbf { \\nabla } \\cdot n$ sentence is usually represented by $n$ embedding vectors, and thus when applying our Greedy algorithm, at each iteration we will try to add each embedding vector to the set $S _ { r }$ and choose the one with largest reward. Since there are only at most $n$ choices, the Greedy algorithm doesn’t suffer much from noise and has similar behavior with Reg-Greedy. ",
+        "bbox": [
+            174,
+            494,
+            825,
+            564
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We perform experiments on an LSTM network which learns to classify a given sentence into one of the ten classes (Society, Science, Health, . . . ). We showcase an example with explanations generated with different methods in Figure 7. We note that although the top-5 relevant keyword sets generated by the three methods do not vary much, the rankings within the highlighted keywords for each explanation are in fact different. We observe that our method Greedy tends to generate explanation that matches human intuition the most. Particularly, to predict the label of “sport”, one might consider “cleats”, “football”, and “cut” as the strongest indications towards the concept “sport”. ",
+        "bbox": [
+            174,
+            570,
+            825,
+            669
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            689,
+            344,
+            705
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our work proposes an objective measurement of feature-based explanation by measuring the “minimum adversarial perturbation” in adversarial literature, which is estimated by adversarial attack. We provide a necessarily incomplete review on related works in objective measurement of explanations and adversarial robustness. ",
+        "bbox": [
+            174,
+            713,
+            823,
+            768
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Objective Measurements for Explanations Evaluation of explanations has been a difficult problem mainly due to the absence of ground truth (Ancona et al., 2018; Sundararajan et al., 2017). Although one could rely on human intuitions to assess the quality of the generated explanations (Lundberg & Lee, 2017; Doshi-Velez & Kim, 2017), for example, judging whether the explanation focuses on the object of interest in an image classification task, these evaluations subject to human perceptions are prone to fall into the pitfall of favoring user-friendly explanations, such as attributions that visually aligns better with the input image, which might not reflect the model behavior (Adebayo et al., 2018). As a result, in addition to subjective measurements, recent literature has also proposed objective measurements, which is also called functionally-grounded evaluations (DoshiVelez & Kim, 2017). We roughly categorize existing objective measurements into two families. ",
+        "bbox": [
+            174,
+            785,
+            825,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This first family of explanation evaluation is called fidelity-based measurement. This includes that Completeness or Sum to Delta which requires the sum of attributions to equal the prediction difference of the original input and baseline (Sundararajan et al., 2017; Shrikumar et al., 2017); sensitivity-n which further generalizes completeness to any subset of the feature (Ancona et al., 2018); local accuracy Ribeiro et al. (2016); Lundberg & Lee (2017); and infidelity which is a framework that encompasses several (Yeh et al., 2019). The general philosophy for this line of methods is to require the sum of attribution value faithfully reflect the change in prediction function value given the presence or absence of certain subset of features. The second family of explanation evaluation are removal-based and preservation-based measurements, which focus on identifying the most important set of features with respect to a particular prediction. The underlying assumption made is that by removing the most (least) salient feature, the resulting function value should drop (increase) the most. (Samek et al., 2016) proposed this idea as an evaluation to evaluate the ranking of featureattribution score. Later on, Fong & Vedaldi (2017) derive explanations by solving an optimization problem to optimize the evaluation. And Dabkowski & Gal (2017) proposed to learn the explanation generating process by training an auxiliary model. ",
+        "bbox": [
+            174,
+            104,
+            825,
+            311
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We note the implicitly in the evaluation process of both fidelity and removal (preservation) based measurement involves computing the change in function value given some set of features being absent. However, it is difficult to carefully model the concept of feature absence in practice, as most models by construction are not able to handle inputs with real missing features. As a result, previous work has compromised by using approximation to estimate the effect of removing certain features. This includes setting the values of the features to be removed by zero (Ancona et al., 2018; Sundararajan et al., 2017) or the mean value (Lundberg & Lee, 2017), blurred value (Fong & Vedaldi, 2017), random value (Samek et al., 2016; Dabkowski & Gal, 2017), or more advanced generative model that attempts to model the given data distribution (Chang et al., 2018). Unfortunately, such approximations that represent feature absence by setting the their values to some predefined distribution would inevitably introduce bias into the evaluation process. With the presence of this inherent caveat, we are thus inspired to adopt another angle to tackle the explanation problem. ",
+        "bbox": [
+            174,
+            319,
+            825,
+            484
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Adversarial Robustness Adversarial robustness has been extensively studied in the past few years. The adversarial robustness of a machine learning model on a given sample can be defined as the shortest distance from the sample to the decision boundary, which corresponds to our definition in (1). Algorithms have been proposed for finding adversarial examples (feasible solutions of (1)), including (Goodfellow et al., 2014; Carlini & Wagner, 2017; Madry et al., 2017). However, those algorithms only work for neural networks, while for other models such as tree based models or nearest neighbor classifiers, adversarial examples can be found by decision based attacks (Brendel et al., 2017; Cheng et al., 2018; Chen et al., 2019). Therefore the proposed framework can also be used in other decision based classifiers. On the other hand, several works aim to solve the neural network verification problem, which is equivalent to finding a lower bound of (1). Examples include (Singh et al., 2018; Wong & Kolter, 2018; Zhang et al., 2018). In principal, our work can also apply these verification methods for getting an approximate solution of (1), but in practice they are very slow to run and often gives loose lower bounds on regular trained networks. ",
+        "bbox": [
+            174,
+            501,
+            825,
+            681
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our work is also closely related to related works that consider the question ”For situation A, why was the outcome B and not $\\mathbf { { C } } ^ { \\ast }$ , which we call counterfactual explanations. Xu et al. (2018) add group sparsity regularization to adversarial attack to enforce semantic structure for the perturbation, which is more interpretable. Ribeiro et al. (2018) find a set of features that once fixed, probability of the prediction is high when perturbing other features. Goyal et al. (2019) show how one could change the input feature such that the system would output a different class, where the change is limited to replacing a part of input feature by a part of an distractor image. Dhurandhar et al. (2018) consider the pertinent negative in a binary setting by solving a carefully designed loss function. ",
+        "bbox": [
+            174,
+            688,
+            825,
+            800
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            820,
+            318,
+            835
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we establish the link between a set of features to a prediction with a new evaluation criteria, robustness analysis, which measures the minimum tolerance of adversarial perturbation. Furthermore, we develop a new explanation method to find important set of features to optimize this new criterion. Experimental results demonstrate that the proposed new explanations are indeed capturing significant feature sets across multiple domains. ",
+        "bbox": [
+            174,
+            851,
+            823,
+            921
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            287,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Julius Adebayo, Justin Gilmer, Michael Muelly, Ian Goodfellow, Moritz Hardt, and Been Kim. Sanity checks for saliency maps. In Advances in Neural Information Processing Systems, pp. 9525–9536, 2018. ",
+        "bbox": [
+            173,
+            126,
+            821,
+            167
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Marco Ancona, Enea Ceolini, Cengiz Oztireli, and Markus Gross. A unified view of gradient-based attribution methods for deep neural networks. International Conference on Learning Representations, 2018. ",
+        "bbox": [
+            173,
+            176,
+            821,
+            219
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sebastian Bach, Alexander Binder, Gre´goire Montavon, Frederick Klauschen, Klaus-Robert Mu¨ller, and Wojciech Samek. On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation. PloS one, 10(7):e0130140, 2015. ",
+        "bbox": [
+            174,
+            228,
+            823,
+            271
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Wieland Brendel, Jonas Rauber, and Matthias Bethge. Decision-based adversarial attacks: Reliable attacks against black-box machine learning models. arXiv preprint arXiv:1712.04248, 2017. ",
+        "bbox": [
+            173,
+            279,
+            825,
+            309
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Nicholas Carlini and David Wagner. Towards evaluating the robustness of neural networks. In 2017 IEEE Symposium on Security and Privacy $( S P )$ , pp. 39–57. IEEE, 2017. ",
+        "bbox": [
+            173,
+            316,
+            823,
+            345
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chun-Hao Chang, Elliot Creager, Andrew A. Goldenberg, and David Kristjanson Duvenaud. Explaining image classifiers by counterfactual generation. In ICLR 2019, 2018. ",
+        "bbox": [
+            173,
+            353,
+            823,
+            383
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Hongge Chen, Huan Zhang, Duane Boning, and Cho-Jui Hsieh. Robust decision trees against adversarial examples. In ICML, 2019. ",
+        "bbox": [
+            171,
+            390,
+            821,
+            420
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Minhao Cheng, Thong Le, Pin-Yu Chen, Jinfeng Yi, Huan Zhang, and Cho-Jui Hsieh. Queryefficient hard-label black-box attack: An optimization-based approach. arXiv preprint arXiv:1807.04457, 2018. ",
+        "bbox": [
+            173,
+            428,
+            821,
+            472
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Piotr Dabkowski and Yarin Gal. Real time image saliency for black box classifiers. In NIPS, 2017. ",
+        "bbox": [
+            171,
+            479,
+            820,
+            496
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Amit Dhurandhar, Pin-Yu Chen, Ronny Luss, Chun-Chen Tu, Paishun Ting, Karthikeyan Shanmugam, and Payel Das. Explanations based on the missing: Towards contrastive explanations with pertinent negatives. In Advances in Neural Information Processing Systems, pp. 592–603, 2018. ",
+        "bbox": [
+            174,
+            502,
+            825,
+            559
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Finale Doshi-Velez and Been Kim. Towards a rigorous science of interpretable machine learning. arXiv preprint arXiv:1702.08608, 2017. ",
+        "bbox": [
+            171,
+            568,
+            823,
+            598
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ruth C. Fong and Andrea Vedaldi. Interpretable explanations of black boxes by meaningful perturbation. 2017 IEEE International Conference on Computer Vision (ICCV), pp. 3449–3457, 2017. ",
+        "bbox": [
+            171,
+            606,
+            823,
+            635
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Timon Gehr, Matthew Mirman, Dana Drachsler-Cohen, Petar Tsankov, Swarat Chaudhuri, and Martin Vechev. Ai2: Safety and robustness certification of neural networks with abstract interpretation. In 2018 IEEE Symposium on Security and Privacy $( S P )$ , pp. 3–18. IEEE, 2018. ",
+        "bbox": [
+            174,
+            642,
+            825,
+            686
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. arXiv preprint arXiv:1412.6572, 2014. ",
+        "bbox": [
+            173,
+            694,
+            823,
+            723
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yash Goyal, Ziyan Wu, Jan Ernst, Dhruv Batra, Devi Parikh, and Stefan Lee. Counterfactual visual explanations. In International Conference on Machine Learning, pp. 2376–2384, 2019. ",
+        "bbox": [
+            174,
+            731,
+            823,
+            761
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Guy Katz, Clark Barrett, David L Dill, Kyle Julian, and Mykel J Kochenderfer. Reluplex: An efficient smt solver for verifying deep neural networks. In International Conference on Computer Aided Verification, pp. 97–117. Springer, 2017. ",
+        "bbox": [
+            173,
+            768,
+            825,
+            813
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Pang Wei Koh and Percy Liang. Understanding black-box predictions via influence functions. In International Conference on Machine Learning, pp. 1885–1894, 2017. ",
+        "bbox": [
+            171,
+            820,
+            823,
+            849
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jiwei Li, Will Monroe, and Dan Jurafsky. Understanding neural networks through representation erasure. CoRR, abs/1612.08220, 2016. URL http://arxiv.org/abs/1612.08220. ",
+        "bbox": [
+            173,
+            858,
+            823,
+            887
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Scott M Lundberg and Su-In Lee. A unified approach to interpreting model predictions. In Advances in Neural Information Processing Systems, pp. 4765–4774, 2017. ",
+        "bbox": [
+            173,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. arXiv preprint arXiv:1706.06083, 2017. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jose Oramas, Kaili Wang, and Tinne Tuytelaars. Visual explanation by interpretation: Improving visual feedback capabilities of deep neural networks. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id $\\underline { { \\underline { { \\mathbf { \\Pi } } } } } =$ H1ziPjC5Fm. ",
+        "bbox": [
+            173,
+            156,
+            825,
+            199
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Vitali Petsiuk, Abir Das, and Kate Saenko. Rise: Randomized input sampling for explanation of black-box models. arXiv preprint arXiv:1806.07421, 2018. ",
+        "bbox": [
+            174,
+            209,
+            823,
+            238
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Gregory Plumb, Denali Molitor, and Ameet S Talwalkar. Model agnostic supervised local explanations. In Advances in Neural Information Processing Systems, pp. 2515–2524, 2018. ",
+        "bbox": [
+            173,
+            248,
+            823,
+            279
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Why should i trust you?: Explaining the predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1135–1144. ACM, 2016. ",
+        "bbox": [
+            173,
+            287,
+            825,
+            332
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Anchors: High-precision model-agnostic explanations. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018. ",
+        "bbox": [
+            171,
+            340,
+            823,
+            371
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hadi Salman, Greg Yang, Huan Zhang, Cho-Jui Hsieh, and Pengchuan Zhang. A convex relaxation barrier to tight robust verification of neural networks. arXiv preprint arXiv:1902.08722, 2019. ",
+        "bbox": [
+            173,
+            380,
+            823,
+            410
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Wojciech Samek, Alexander Binder, Gre´goire Montavon, Sebastian Lapuschkin, and Klaus-Robert Mu¨ller. Evaluating the visualization of what a deep neural network has learned. IEEE transactions on neural networks and learning systems, 28(11):2660–2673, 2016. ",
+        "bbox": [
+            174,
+            419,
+            825,
+            463
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Avanti Shrikumar, Peyton Greenside, and Anshul Kundaje. Learning important features through propagating activation differences. International Conference on Machine Learning, 2017. ",
+        "bbox": [
+            173,
+            472,
+            823,
+            502
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Gagandeep Singh, Timon Gehr, Matthew Mirman, Markus Pu¨schel, and Martin Vechev. Fast and effective robustness certification. In Advances in Neural Information Processing Systems, pp. 10802–10813, 2018. ",
+        "bbox": [
+            173,
+            511,
+            823,
+            554
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. In International Conference on Machine Learning, 2017. ",
+        "bbox": [
+            169,
+            564,
+            823,
+            594
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Shiqi Wang, Kexin Pei, Justin Whitehouse, Junfeng Yang, and Suman Jana. Efficient formal safety analysis of neural networks. In Advances in Neural Information Processing Systems, pp. 6367– 6377, 2018. ",
+        "bbox": [
+            173,
+            603,
+            823,
+            647
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tsui-Wei Weng, Huan Zhang, Hongge Chen, Zhao Song, Cho-Jui Hsieh, Luca Daniel, Duane Boning, and Inderjit Dhillon. Towards fast computation of certified robustness for relu networks. In International Conference on Machine Learning, pp. 5273–5282, 2018a. ",
+        "bbox": [
+            174,
+            656,
+            825,
+            700
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tsui-Wei Weng, Huan Zhang, Pin-Yu Chen, Jinfeng Yi, Dong Su, Yupeng Gao, Cho-Jui Hsieh, and Luca Daniel. Evaluating the robustness of neural networks: An extreme value theory approach. arXiv preprint arXiv:1801.10578, 2018b. ",
+        "bbox": [
+            174,
+            710,
+            826,
+            753
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Eric Wong and Zico Kolter. Provable defenses against adversarial examples via the convex outer adversarial polytope. In International Conference on Machine Learning, pp. 5283–5292, 2018. ",
+        "bbox": [
+            173,
+            763,
+            825,
+            792
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Kaidi Xu, Sijia Liu, Pu Zhao, Pin-Yu Chen, Huan Zhang, Quanfu Fan, Deniz Erdogmus, Yanzhi Wang, and Xue Lin. Structured adversarial attack: Towards general implementation and better interpretability. arXiv preprint arXiv:1808.01664, 2018. ",
+        "bbox": [
+            176,
+            803,
+            823,
+            845
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Chih-Kuan Yeh, Cheng-Yu Hsieh, Arun Sai Suggala, David I. Inouye, and Pradeep Ravikumar. On the (in)fidelity and sensitivity for explanations. CoRR, abs/1901.09392, 2019. ",
+        "bbox": [
+            171,
+            856,
+            823,
+            885
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pp. 818–833. Springer, 2014. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Huan Zhang, Tsui-Wei Weng, Pin-Yu Chen, Cho-Jui Hsieh, and Luca Daniel. Efficient neural network robustness certification with general activation functions. In Advances in neural information processing systems, pp. 4939–4948, 2018. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Huan Zhang, Pengchuan Zhang, and Cho-Jui Hsieh. Recurjac: An efficient recursive algorithm for bounding jacobian matrix of neural networks and its applications. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 5757–5764, 2019. ",
+        "bbox": [
+            176,
+            155,
+            823,
+            196
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, and Kai-Wei Chang. Men also like shopping: Reducing gender bias amplification using corpus-level constraints. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pp. 2979–2989, 2017. ",
+        "bbox": [
+            174,
+            207,
+            826,
+            262
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A EVALUATION CURVES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            393,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/a47860d5bb1b45e69b2d70e32d53f2d9373bbd412d6d920222f16b83980ea971.jpg",
+        "image_caption": [
+            "Figure 8: Comparisons between our proposed methods under different criteria. From left to right: untargeted Robustness- $\\overline { { \\boldsymbol { S } _ { r } } }$ , targeted Robustness- $\\overline { { \\boldsymbol { S } _ { r } } }$ , untargeted Robustness- $S _ { r }$ , targeted Robustness$S _ { r }$ . We omit points in the plot with value too high to fit in the scale of y-axis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            147,
+            810,
+            234
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/5727086bf61d07de10d03f21470c816f1a5373474f0ae8d8ee1a8a41cf50af51.jpg",
+        "image_caption": [
+            "Figure 9: Comparisons between our proposed methods and existing explanations under different criteria. From left to right: untargeted Robustness- $\\overline { { \\boldsymbol { S } _ { r } } }$ , targeted Robustness- $\\overline { { S _ { r } } }$ , untargeted Robustness$S _ { r }$ , targeted Robustness- $S _ { r }$ . We omit points in the plot with value too high to fit in the scale of y-axis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            316,
+            810,
+            404
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/a9350862af1406cdbf707a1ca40e6693eccfed8ac8ff4da0f9bb839c321ebe2c.jpg",
+        "image_caption": [
+            "Figure 10: Comparisons between our proposed methods under different criteria on ImageNet. From left to right: untargeted Robustness- $\\overline { { \\boldsymbol { S } _ { r } } }$ , targeted Robustness- $\\overline { { \\boldsymbol { S } _ { r } } }$ , untargeted Robustness- $S _ { r }$ , targeted Robustness- $S _ { r }$ . We omit points in the plot with value too high to fit in the scale of y-axis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            500,
+            810,
+            587
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/5666adba4deefe3b02d28f9206b3f84130a620acfba171f8c8c6a60f558f8b96.jpg",
+        "image_caption": [
+            "Figure 11: Comparisons between our proposed methods under different criteria on ImageNet. From left to right: untargeted Robustness- $\\vec { \\cdot S _ { r } }$ , targeted Robustness- $\\overline { { \\boldsymbol { S } _ { r } } }$ , untargeted Robustness- $S _ { r }$ , targeted Robustness- $S _ { r }$ . We omit points in the plot with value too high to fit in the scale of y-axis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            669,
+            810,
+            756
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/a925714ae0ea3b8327a809c6052b8e0494eefd2f6cc6eedef6f86b6406c0636a.jpg",
+        "image_caption": [
+            "Figure 12: Comparisons between explanations under different criteria on MNIST. Left figure: change in output logits as relevant features are inserted into the input. Right figure: change in output logits as relevant features are removed from the input. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            154,
+            741,
+            290
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/991ba8ae20919ccaa941a9ccb4ecafc91235dce72db5ded41a4cf71d6b803fad.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LOO</td><td>BBMP</td><td>Reg-Greedy</td></tr><tr><td>Insertion</td><td>250.81</td><td>262.74</td><td>200.50</td><td>192.44</td><td>102.53</td><td>379.15</td></tr><tr><td>Deletion</td><td>281.88</td><td>273.71</td><td>362.68</td><td>442.65</td><td>527.80</td><td>159.77</td></tr></table>",
+        "bbox": [
+            256,
+            363,
+            738,
+            417
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Area under curve of the Insertion and Deletion criteria for various explanations on MNIST.   \nThe higher the better for Insertion; the lower the better for Deletion. ",
+        "bbox": [
+            171,
+            428,
+            823,
+            455
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C T-TEST ON AUC ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            484,
+            346,
+            501
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/b9e78df70808b8df55365114347a344e34482978fb869af618dfe8d58255ce81.jpg",
+        "table_caption": [
+            "Table 6: The proposed Reg-Greedy versus other explanations on MNIST under our proposed criteria with Student’s $t$ -test at $9 5 \\%$ confidence level. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LOO</td><td>BBMP</td></tr><tr><td>Robustness-Sr</td><td>win</td><td>win</td><td>win</td><td>win</td><td>win</td></tr><tr><td>Robustness-Sr</td><td>win</td><td>win</td><td>win</td><td>win</td><td>win</td></tr></table>",
+        "bbox": [
+            318,
+            521,
+            678,
+            577
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/8c7193f111bc1689cc6aa06277d0153a7bba519770f2c0dbcc8999717515c36a.jpg",
+        "table_caption": [
+            "Table 7: The proposed Reg-Greedy versus other explanations on MNIST under Insertion and Deletion criteria with Student’s $t$ -test at $9 5 \\%$ confidence level. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LO0</td><td>BBMP</td></tr><tr><td>Insertion</td><td>win</td><td>win</td><td>win</td><td>win</td><td>win</td></tr><tr><td>Deletion</td><td>win</td><td>win</td><td>win</td><td>win</td><td>win</td></tr></table>",
+        "bbox": [
+            325,
+            632,
+            673,
+            688
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D SANITY CHECK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            753,
+            341,
+            770
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/114ad598cab2ba92945aa7ef2d7394a8f6c9dbb11804038fbf2e303b49c1508b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Explanations</td><td>Grad</td><td>IG</td><td>SHAP</td><td>LOO</td><td>BBMP</td><td>Reg-Greedy</td></tr><tr><td>Rank Correlation</td><td>0.3001</td><td>0.3042</td><td>0.1108</td><td>0.4966</td><td>0.1775</td><td>0.1835</td></tr></table>",
+        "bbox": [
+            243,
+            789,
+            750,
+            832
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 8: Rank correlation between explanations with respect to original and randomized model. ",
+        "bbox": [
+            184,
+            840,
+            808,
+            856
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E COMPARISONS ON TARGETED EXPLANATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            583,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/051a251c11e28f15ca92221d21bf76f0f346a9a48a57ecbfefc10c81a8ea8634.jpg",
+        "image_caption": [
+            "Figure 13: Comparisons between different targeted explanations against different targeted class on MNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            137,
+            823,
+            267
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "F HEATMAP VISUALIZATION ON MNIST ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            333,
+            529,
+            349
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/85b4204194e7049b8f7a2b9e33fb6a6fc348a91de3e493ae5d47e24b19e67520.jpg",
+        "image_caption": [
+            "Figure 14: Heatmap Visualization of different explanations on MNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            364,
+            797,
+            882
+        ],
+        "page_idx": 15
+    }
+]

parse/train/S1eYKlrYvr/S1eYKlrYvr_content_list.json

@@ -0,0 +1,1599 @@
+[
+    {
+        "type": "text",
+        "text": "DIAGNOSING THE ENVIRONMENT BIAS IN VISION-AND-LANGUAGE NAVIGATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            679,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Vision-and-Language Navigation (VLN) requires an agent to follow naturallanguage instructions, explore the given environments, and reach the desired target locations. These step-by-step navigational instructions are extremely useful in navigating new environments that the agent does not know about previously. Most recent works that study VLN observe a significant performance drop when tested on unseen environments (i.e., environments not used in training), indicating that the neural agent models are highly biased towards training environments. Although this issue is considered as one of the major challenges in VLN research, it is still under-studied and needs a clearer explanation. In this work, we design novel diagnosis experiments via environment re-splitting and feature replacement, looking into possible reasons for this environment bias. We observe that neither the language nor the underlying navigational graph, but the low-level visual appearance conveyed by ResNet features directly affects the agent model and contributes to this environment bias in results. According to this observation, we explore several kinds of semantic representations which contain less low-level visual information, hence the agent learned with these features could be better generalized to unseen testing environments. Without modifying the baseline agent model and its training method, our explored semantic features significantly decrease the performance gap between seen and unseen on multiple datasets (i.e., $8 . 6 \\%$ to $0 . 2 \\%$ on R2R, $2 3 . 9 \\%$ to $0 . 1 \\%$ on R4R, and 3.74 to 0.17 on CVDN) and achieve competitive unseen results to previous state-of-the-art models. ",
+        "bbox": [
+            233,
+            263,
+            764,
+            555
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            580,
+            334,
+            595
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Vision-and-Language Navigation (VLN) tests an agent’s ability to follow complex natural language instructions as well as explore the given environments, so as to be able to reach the desired target locations. As shown in Fig. 1, the agent is put in an environment and given a detailed step-by-step navigational instruction. With these inputs, the agent needs to navigate the environment and find the correct path to the target location. In this work, we focus on the instruction-guided navigation (MacMahon et al., 2006; Anderson et al., 2018b; Misra et al., 2018; Blukis et al., 2018; Chen et al., 2019c) where detailed step-by-step navigational instructions are used (e.g., ‘Go outside the dining room and turn left ...’), in contrast to the target-oriented navigation (Gordon et al., 2018; Das et al., 2018; Mirowski et al., 2018; Yu et al., 2019) where only the target is referred (e.g., ‘Go to the kitchen’ or ‘Tell me the color of the bedroom’). Although these step-by-step instructions are overdetailed when navigating local areas (e.g., your home), they are actively used in unseen environments (e.g., your friend’s house, a new city) where the desired target is usually unknown to navigational agents. For this purpose, testing on unseen environments which are not used during agent-training is important and widely accepted by instruction-guided navigation datasets. ",
+        "bbox": [
+            174,
+            611,
+            825,
+            805
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent works propose different methods to improve generalizability of agents on these unseen testing environments; and most of the existing works (Anderson et al., 2018b; Wang et al., 2018b; Fried et al., 2018; Wang et al., 2019b; Ma et al., 2019a;b; Tan et al., 2019; Huang et al., 2019; Hu et al., 2019) observe a significant performance drop from seen environments (i.e., the environments used in training) to unseen environments (i.e., the environments not used in training), which indicates a strong bias in the model towards the training environments. While this performance gap is emphasized as one of the major challenges in current VLN research, the issue is still left unresolved and waits for an explicit explanation. Thus, in this paper, we aim to answer three questions to this environment bias: 1. Where (i.e., in which component) is the bias located? 2. Why does this bias exist? 3. How to eliminate this bias? ",
+        "bbox": [
+            174,
+            813,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/0f93e130663742430e83365ba4e9f0ff9e6567b8ab35e7d864d7ce32ccfd97a3.jpg",
+        "image_caption": [
+            "Figure 1: Vision-language-navigation: performance of the agent drops in unseen environments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            223,
+            99,
+            771,
+            286
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            323,
+            820,
+            351
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To locate where the bias is, we start by showing that natural-language navigational instructions and underlying navigational graphs are not direct reasons for this performance gap. We then investigate the effect of environments on the agent’s performance. In order to conduct a detailed analysis, we resplit the environment and categorize the validation data into three sets based on their visibility to the training set: path-seen data intersecting with the training paths, path-unseen data using the training environments but away from the training paths, and env-unseen data using unseen environments (environments not used in training). By showing that the results gradually decrease from path-seen data to env-unseen data, we characterize the environment bias at three levels: path level, region level, and environment level. ",
+        "bbox": [
+            174,
+            358,
+            825,
+            483
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "These three levels of environment biases indicate strong ‘spatial localities’ in the tasks of VLN, which are intuitively reasonable because environments and regions (e.g., houses and cities) usually have their own styles when built or decorated. We next want to analyze the detailed reason why this locality would further lead to a gap in seen versus unseen results. Our hypothesis is that the low-level information carried by the ResNet features (He et al., 2016) is the reason. To keep minimal low-level visual information and promote more high-level semantic information, we replace the ResNet features with the 1000 ImageNet classification probabilities. Although the semantic information encoded by these features is not accurate because of the shifted domain of images and labels, the same model with ImageNet-Labels features performs surprisingly well on various VLN datasets (i.e., Room-to-Room, R4R, and $\\mathrm { C V D N ^ { 1 } }$ ). Most importantly, these noisy semantic features effectively eliminate the performance gap between seen and unseen environments, which suggests that the environment bias is attributed to the ResNet features as our hypothesis. ",
+        "bbox": [
+            173,
+            491,
+            825,
+            656
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Following the practice in using ImageNet labels as semantic features, we further provide a discussion on how the environment bias could be eliminated. For this, we employ advanced high-level semantic features which are more rational for the VLN domain. We explore three kinds of semantic features: (1) areas of detected object labels (Ren et al., 2015); (2) ground truth semantic views (Chang et al., 2017); and (3) learned semantic view features. We show that all of these semantic features significantly reduce the environment bias in multiple datasets and also achieve strong results in testing unseen environments. We hope this work encourages more investigation and research into improving the generalization of vision-language models to unseen real-world scenarios. ",
+        "bbox": [
+            174,
+            664,
+            825,
+            775
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            796,
+            344,
+            813
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Vision-and-Language Navigation: Vision-and-language navigation is an emerging task in the vision-and-language area. A lot of datasets have been proposed in recent years, such as Roomto-Room (Anderson et al., 2018b), Room-for-Room (Jain et al., 2019), TouchDown (Chen et al., 2019c), CVDN (Thomason et al., 2019b), RERERE (Qi et al., 2019), House3D (Wu et al., 2018) and EQA (Das et al., 2018). Recent works (Thomason et al., 2019a; Wang et al., 2018b; Fried et al., ",
+        "bbox": [
+            174,
+            828,
+            823,
+            898
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2018; Wang et al., 2019b; Ma et al., 2019a;b; Tan et al., 2019; Hu et al., 2019; Ke et al., 2019; Anderson et al., 2019) focusing on improving the performance of navigation models, especially in unseen testing environments, have helped to increase the navigational success rate. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Domain Adaptation: The general setup of domain adaption contains two sets of data samples $\\{ x _ { i } \\} _ { x _ { i } \\in X }$ and $\\{ y _ { i } \\} _ { y _ { i } \\in Y }$ from two domains $X$ and $Y$ . Based on these samples, we could learn domain invariant feature with adversarial training (Goodfellow et al., 2014; Zhu et al., 2017; Long et al., 2018; Wang et al., 2019a; Hosseini-Asl et al., 2019; Zhang et al., 2019; Gong et al., 2019; Chen et al., 2019b) or learn a transfer function $f : X \\to Y$ (Wang et al., 2018a; Chen et al., 2019a; Rozantsev et al., 2018). However, samples from the target domain may not be available (e.g., the testing environments in navigation should not be used in training) in applications. Thus, we try to give an interpretable explanation to why performance varies in different domains and design a robust feature for it without deliberately considering the target domain. Two methods in VLN, RCM (Wang et al., 2019b) and EnvDrop (Tan et al., 2019), explore the possibility of domain adaptation. Both works take the testing environments in training while RCM also uses testing instructions. ",
+        "bbox": [
+            174,
+            154,
+            825,
+            305
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Domain Generalization: In domain generalization (Blanchard et al., 2011), the goal is to predict the labels in the previous unseen domain. Similar to the test setting of VLN tasks, the testing data is unrevealed in training. Works have been proposed to learn the common features of the training domain (Muandet et al., 2013; Blanchard et al., 2017; Li et al., 2017; 2018; Carlucci et al., 2019; Deshmukh et al., 2019). In this paper, we focus on the domain generalization problem in VLN task, and try to find the reasons for the failures. ",
+        "bbox": [
+            174,
+            313,
+            825,
+            396
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 VISION-AND-LANGUAGE NAVIGATION AND ITS ENVIRONMENT BIAS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            417,
+            779,
+            434
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We first introduce the task of vision-and-language navigation (VLN) and briefly describe the neural agent models used in our work. We next survey previous works on multiple indoor navigation datasets to show that the environment bias is widely observed in current VLN research. Lastly, we claim that this bias also exists in the outdoor navigation tasks, if the agent is tested on unseen regions. ",
+        "bbox": [
+            174,
+            450,
+            825,
+            520
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 VISION-AND-LANGUAGE NAVIGATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            539,
+            482,
+            553
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Tasks: As shown in Fig. 1, the goal of the VLN task is to train an agent to navigate a certain type of environments $\\{ { \\bf E } \\}$ (e.g., indoor or outdoor environments) given the instruction I. Each environment $\\mathbf { E }$ is an independent space, such as a room or a house, and consists of a set of viewpoints. Each viewpoint is represented as a panoramic image and can be decomposed into separate views $\\{ o \\}$ as inputs to the neural agent models. The viewpoints and their connectivity form the navigational graph. In practice, after being placed at a particular viewpoint and given the instruction in the beginning, at each time step, the agent can observe the panoramic image of the viewpoint where it is located, and choose to move along an edge of the graph to the next node (i.e., viewpoint) or stop. This navigational process produces a path (i.e., a list of viewpoints), and the performance of the agent is evaluated by whether it reaches the target location that the instruction indicates in the end. ",
+        "bbox": [
+            173,
+            565,
+            825,
+            704
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Neural Agent Models: Most instruction-guided navigational agents are built based on attentive encoder-decoder models (Bahdanau et al., 2015). The encoder reads the instructions while the decoder outputs actions based on the encoded instructions and perceived environments. Since the main purpose of this work is to understand the environment bias in vision-and-language navigation, we use a minimal representative neural agent model that achieves comparable results to previous works. Specifically, we adopt the panoramic-view neural agent model in Fried et al. (2018) (‘Follower’) with modifications from Tan et al. (2019) as our baseline model. We also exclude advanced training techniques (i.e., reinforcement learning and data augmentation) and only train the agent with imitation learning in all our experiments for the same purpose. More details in original papers. ",
+        "bbox": [
+            174,
+            710,
+            825,
+            837
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 ENVIRONMENT BIAS IN INDOOR NAVIGATION ",
+        "text_level": 1,
+        "bbox": [
+            179,
+            854,
+            529,
+            869
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In order to evaluate the generalizability of agent models, indoor vision-and-language navigation datasets (e.g., those collected from Matterport3D (Chang et al., 2017)) use disjoint sets of environments in training and testing. Most of the datasets provide two validation splits to verify the agent’s performance in both sets of environments: validation seen, which takes the data from training environments, and validation unseen, whose data is from new environments apart from the training environments. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/46ca292073ba8977b66b5c6eb34a303a26fa18ac31d6f53c246d0de9710256dd.jpg",
+        "table_caption": [
+            "Table 1: Results show the performance gap between seen (‘Val Seen’) and unseen (‘Val Unseen’) environments in several VLN tasks. Room-to-Room and Room-for-Room are evaluated with ‘Success Rate’, CVDN is evaluated with ‘Goal Progress’, Touchdown is evaluated with ‘Task Completion’. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Task</td><td rowspan=\"2\">Method</td><td colspan=\"3\">Result</td></tr><tr><td>Val Seen</td><td>Val Unseen</td><td>Abs Gap |△|</td></tr><tr><td rowspan=\"10\">Room-to-Room (Anderson et al.,2018b)</td><td>R2R (Anderson et al.,2018b)</td><td>38.6</td><td>21.8</td><td>16.8</td></tr><tr><td>RPA (Wang et al., 2018b)</td><td>42.9</td><td>24.6</td><td>18.3</td></tr><tr><td>S-Follower (Fried et al., 2018)</td><td>66.4</td><td>35.5</td><td>30.9</td></tr><tr><td>RCM(Wang et al.,2019b)</td><td>66.7</td><td>42.8</td><td>23.9</td></tr><tr><td>SMNA (Ma et al., 2019a)</td><td>67</td><td>45</td><td>22</td></tr><tr><td>Regretful (Ma et al.,2019b)</td><td>69</td><td>50</td><td>19</td></tr><tr><td>EnvDrop (Tan et al., 2019)</td><td>62.1</td><td>52.2</td><td>9.9</td></tr><tr><td>ALTR (Huang et al.,2019)</td><td>55.8</td><td>46.1</td><td>9.7</td></tr><tr><td>RN+Obj (Hu et al., 2019)</td><td>59.2</td><td>39.5</td><td>19.7</td></tr><tr><td>CG (Anderson et al., 2019) Our baseline</td><td>31</td><td>31</td><td>0</td></tr><tr><td>Our learned-semantic</td><td></td><td>56.1 53.1</td><td>47.5</td><td>8.6</td></tr><tr><td rowspan=\"4\">Room-for-Room (Jain et al., 2019)</td><td></td><td></td><td>53.3</td><td>0.2</td></tr><tr><td>Speaker-Follower</td><td>51.9</td><td>23.8</td><td>28.1</td></tr><tr><td>RCM</td><td>55.5</td><td>28.6</td><td>26.9</td></tr><tr><td>Our baseline Our learned-semantic</td><td>54.6 36.2</td><td>30.7 36.1</td><td>23.9</td></tr><tr><td rowspan=\"3\">CVDN (Thomason et al., 2019b)</td><td>NDH</td><td>5.92</td><td>2.10</td><td>0.1</td></tr><tr><td>Our baseline</td><td>5.97</td><td>2.23</td><td>3.82 3.74</td></tr><tr><td>Our learned-semantic</td><td>2.60</td><td>2.43</td><td>0.17</td></tr><tr><td rowspan=\"4\">Touchdown (Chen et al., 2019c)</td><td></td><td>7.9 (dev)</td><td></td><td></td></tr><tr><td>GA (original split) RCONCAT (original split)</td><td>9.8 (dev)</td><td>5.5 (test) 10.7 (test)</td><td>1</td></tr><tr><td>Our baseline (original split)</td><td>15.0 (dev)</td><td>14.2 (test)</td><td>1</td></tr><tr><td>Our baseline (seen/unseen split)</td><td>17.5</td><td>5.3</td><td>1 12.2</td></tr></table>",
+        "bbox": [
+            174,
+            150,
+            825,
+            497
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            513,
+            825,
+            555
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the first part of Table 1, we list most of the previous works on the Room-to-Room dataset (Anderson et al., 2018b) and report the success rate under greedy decoding (i.e., without beam-search) on validation seen and validation unseen splits. The large absolute gaps (from $3 0 . 9 \\%$ to $9 . 7 \\%$ ) between the results of seen and unseen environments show that current neural agent models on R2R suffer from environment bias2. Besides Room-to-Room (R2R), we also analyze two newly-released indoor navigation datasets that were also collected from Matterport3D environments: Room-forRoom (R4R) (Jain et al., 2019) and Cooperative Vision-and-Dialog Navigation (CVDN) (Thomason et al., 2019b). As shown in the second and third parts of Table. 1, results drop significantly from seen to unseen environments (i.e., $2 6 . 9 \\%$ on R4R and 3.74 on CVDN), indicating that agent models also suffer from the environment bias in these datasets. Lastly, we show the results (denoted as ‘ours’ in Table. 1) when the environment bias (reason analyzed in Sec. 5) is effectively eliminated by our learned semantic features (described in Sec. 6.3). As a result, the performance gaps are effectively decreased on all three datasets without changing the model and learning hyper-parameters, compared to our baselines (denoted as ‘Our baseline’) and previous works 3. ",
+        "bbox": [
+            173,
+            563,
+            825,
+            756
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 ENVIRONMENT BIAS IN OUTDOOR NAVIGATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            776,
+            549,
+            791
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since the three indoor navigational datasets in previous sections are collected from the Matterport3D environments (Chang et al., 2017), in order to show that the environment bias is a general phenomenon also existing in other kinds of environments, we investigate the outdoor navigation task from Touchdown dataset (Chen et al., 2019c), whose environments are taken from New York City. In the original data splits of Touchdown, the environment is not specifically divided into seen and unseen and only involved one city. Thus the trained agent is only tested on the training environments (similar to validation seen split). To reveal the environment bias in Touchdown dataset, we split the city environment according to latitude and create two sub-environments: ‘training’ and ‘unseen’. The data are then re-split into training, val-seen, and val-unseen, accordingly. We adapt our baseline R2R agent model with additional convolutional layers to fit this new task. As shown in the last part of Table. 1, when experimenting on the original data split, our baseline model achieves state-of-theart results on the original ‘dev’ set and ‘test’ set, proving the validity of our model in this dataset. However, the results on our re-split data (denoted as ‘Our baseline (seen/unseen split)’) still show a big drop from the ’training’ to the ’unseen’ sub-environment (from $1 7 . 5 \\%$ to $5 . 3 \\%$ ), indicating that environment bias is a broad issue. ",
+        "bbox": [
+            173,
+            803,
+            823,
+            832
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/1182dd37143299aa3930ca49723be9c5056035f01d331ca93c462899b4d835d5.jpg",
+        "image_caption": [
+            "Figure 2: The language ’distance’ distribution (defined by language scores) and its relationship to success rate. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            276,
+            104,
+            717,
+            345
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            414,
+            825,
+            593
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 WHERE: THE EFFECT OF DIFFERENT TASK COMPONENTS",
+        "text_level": 1,
+        "bbox": [
+            174,
+            622,
+            683,
+            640
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Sec. 3, we showed that current neural agent models are biased towards the training environments on multiple vision-and-language navigation (VLN) datasets. In this section, our goal is to locate the component of VLN tasks which this environment bias is attributed to. As one of the early-released and well-explored datasets of VLN, Room-to-Room (R2R) dataset (Anderson et al., 2018b) is used as the diagnosing dataset in the experiments. We start by showing that two possible candidates, the natural language instructions and the underlying navigational graph, do not directly contribute to the environment bias. Then the effect of visual environments is analyzed in detail. ",
+        "bbox": [
+            174,
+            660,
+            825,
+            757
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 THE EFFECT OF NATURAL-LANGUAGE NAVIGATIONAL INSTRUCTIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            782,
+            700,
+            797
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A common hypothesis is that the navigational instructions for unseen environments (e.g., val unseen) are much different from the training environments (i.e., training and val seen) due to the different objects and layouts in new environments; and this lingual difference thus leads to the performance gap. In this section, we analyze the distributions of success rate with regard to the relationship between validation data’s instructions and training instructions. In order to quantitatively evaluate this relationship, we define the ‘distances’ from a validating instruction to all training instructions as the phrase-matching metric. Suppose $x$ is a validating datum, $\\mathbb { T }$ is the training set, and $\\operatorname { i n s t } ( x )$ is the instruction of the datum $x$ , we use ROUGE-L (Lin, 2004) and BLEU-4 (Papineni et al., 2002) to ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/2d21e90dbdc29d400646db803bfb2541045038f1ecb3bac90bbe6312a9ba0d9e.jpg",
+        "image_caption": [
+            "Figure 3: Graph split: left is original data and right is re-splitting data. Black vertices are viewpoints visited during training; red paths are val seen $/$ val path-seen; blue paths are val path-unseen. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            264,
+            103,
+            732,
+            289
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "calculate this ‘distance’: ",
+        "bbox": [
+            174,
+            349,
+            333,
+            363
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0f6492ec3f2e80fe0b207a59aaabe10d1335687c1656ac9883c2d6994e303939.jpg",
+        "text": "$$\n\\operatorname { d i s } _ { \\mathrm { R o U G E } } ( x , \\mathbb { T } ) = \\operatorname* { m i n } _ { t \\in \\mathbb { T } } \\mathrm { R O U G E - L } \\left( \\operatorname { i n s t } ( x ) , \\operatorname { i n s t } ( t ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            369,
+            671,
+            393
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e62817efdf19c3211babc088b533067eab109d3e3fe7c5a0153c2e9b9ae1f21.jpg",
+        "text": "$$\n\\mathrm { d i s } _ { \\mathrm { B L E U } } ( x , \\mathbb { T } ) = \\mathrm { B L E U } \\ – 4 \\left( \\mathrm { i n s t } ( x ) , \\{ \\mathrm { i n s t } ( t ) \\} _ { t \\in \\mathbb { T } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            396,
+            658,
+            415
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where we consider all the training instructions as references in calculating the BLEU-4 score. ",
+        "bbox": [
+            174,
+            419,
+            782,
+            434
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We show the distributions of success rates and distances in Fig. 2. As opposed to the hypothesis, we do not observe a significant difference between the distributions of ‘distances’ (as shown in Fig. 2 (a, b)) on seen validation and unseen validation. For the success rate distributions (in Fig. 2(c,d)), the performance is better on instructions with smaller ‘distances’ (i.e., higher BLEU-4/ROUGE-L scores w.r.t. the training instructions) on both validation splits. However, comparing two splits, with the same ‘distance’ to training instructions, seen validation still significantly outperforms the unseen validation set on success rate, which implies the existence of other reasons rather than language attributed to this performance gap. ",
+        "bbox": [
+            174,
+            440,
+            825,
+            553
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 THE EFFECT OF UNDERLYING NAVIGATIONAL GRAPH ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            568,
+            589,
+            583
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As shown in Fig. 3, an environment could be considered as its underlying navigational graph with visual information (as in Fig. 1). In order to test whether the agent model could overfit to these navigational graphs (and thus be biased towards training environments), we follow the experiments in Hu et al. (2019) to train the agent without visual information. Specifically, we mask out the ResNet features with zero vectors thus the agent could only make the decision based on the instructions and the navigational graph. With our baseline model, the success rate is $3 8 . 5 \\%$ on validation seen and $4 1 . 0 \\%$ on validation unseen in this setting, which is consistent with the finding in Hu et al. (2019). Besides showing the relatively good performance of unseen split without visual contents (similar to Thomason et al. (2019a) and Hu et al. (2019)), we also want to emphasize the low performance gap between seen and unseen environments $2 . 5 \\%$ compared to the $\\bar { > } 1 0 \\%$ gap in usual). Hence, we claim that the underlying graph is not a dominant reason for the environment bias. ",
+        "bbox": [
+            173,
+            594,
+            825,
+            747
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 THE EFFECT OF VISUAL ENVIRONMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            765,
+            500,
+            779
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To show how the visual environments affect the agent’s performance, we analyze the results on unseen environments and in different spatial regions of the training environments. In order to give a detailed characterization of the effect of environments, we are going to reveal the spatial localities which are related to the agent’s performance at three different levels: ",
+        "bbox": [
+            176,
+            790,
+            825,
+            847
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Path-level Locality: Agents are better at paths which intersect with the training paths. • Region-level Locality: Agents are better in regions which are closer to the training data. • Environment-level Locality: Agents perform better on training environments than on unseen environments. ",
+        "bbox": [
+            214,
+            858,
+            823,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/ba847142b3943a732d68be99141c09a591b744b3c478afd73570f623b4730059.jpg",
+        "table_caption": [
+            "Table 2: Results on our re-splitting data showing the path-level and environment-level localities. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Splitting Method</td><td rowspan=\"2\">Train</td><td colspan=\"3\">Validation</td></tr><tr><td></td><td></td><td>Path-seen</td><td>Path-unseen</td><td>Env-unseen</td></tr><tr><td rowspan=\"3\">Environments</td><td>R2R</td><td>61</td><td>56</td><td>0</td><td>11</td></tr><tr><td>X-split</td><td>61</td><td>57</td><td>16</td><td>11</td></tr><tr><td>Z-split</td><td>61</td><td>56</td><td>29</td><td>11</td></tr><tr><td rowspan=\"3\">Number of Data</td><td>R2R</td><td>14,025</td><td>1,020</td><td>0</td><td>2,349</td></tr><tr><td>X-split</td><td>11,631</td><td>1,230</td><td>1,098</td><td>2,349</td></tr><tr><td>Z-split</td><td>10,894</td><td>867</td><td>2,324</td><td>2.349</td></tr><tr><td rowspan=\"3\">Success Rate</td><td>R2R</td><td>88.3</td><td>56.1</td><td>1</td><td>47.5</td></tr><tr><td>X-split</td><td>87.3</td><td>58.9</td><td>52.6</td><td>46.7</td></tr><tr><td>Z-split</td><td>94.7</td><td>62.5</td><td>47.8</td><td>42.4</td></tr></table>",
+        "bbox": [
+            207,
+            121,
+            789,
+            287
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "And the existence of these spatial locality inspires us to find the direct cause of the problem in Sec. 3.2. However, the original split of data is not fine-grained enough to separately reveal these spatial localities. To better illustrate this, we visualize the data from one environment of the Roomto-Room dataset in Fig. 3, where the vertices are viewpoints with visual information and edges are valid connections between viewpoints. The vertices highlighted with dark-black indicate the viewpoints which are used in training paths, and the red edges are the connections covered by original val-seen paths. As shown in Fig. 3, nearly all viewpoints in val-seen paths (vertices connected to red lines) are used as viewpoints in training data (vertices marked by dark-black). We thus cannot categorize the path-level and region-level localities. To bypass this, we propose a novel re-splitting method to create our diagnosis data splits. ",
+        "bbox": [
+            174,
+            310,
+            825,
+            449
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Structural Data Re-splitting We employ two kinds of structural data splitting methods based on the horizontal or vertical coordinates, denoted as $\\mathbf { \\epsilon } ^ { \\bullet } \\mathbf { X }$ -split’ and $^ { \\bullet } \\mathrm { Z }$ -split’, respectively. The $^ { 6 } \\mathrm { Z }$ - split’ intuitively separates different floors in the houses and $\\mathbf { \\epsilon } ^ { \\bullet } \\mathbf { X }$ -split’ creates separate areas. When applying to the training environments in R2R dataset, we use one side of the splitting line (see the ‘X-splitting line’ Fig. 3) as the new training ‘environment’, and the other side as the path-unseen ‘environment’. In addition to this split of environments, we also re-split the original training data and val-seen data while keeping the val-unseen data the same. The data paths across the splitting line are dropped. As shown in the right part of Fig. 3, we create three new data splits: training split, val-path-seen split, and val-path-unseen split. The edges covered by the new val-path-unseen split are highlighted in blue, while the color style of training split and val-path-seen split (‘Black’ for viewpoints in training and ‘Red’ for edges in val path-seen) are the same. Since the amount of original val-seen data are inadequate to fill two new validation sets (val path-seen and val pathunseen), we bring some (original) training data into our new validation splits. The overall statistics of original splits and our new splits are shown in Table 2.4 ",
+        "bbox": [
+            174,
+            465,
+            825,
+            659
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Existence of Path-level and Environment-level Localities For both splitting methods, we train our baseline model on the newly-split training set and evaluate on our three validation sets (denoted as $\\mathbf { \\epsilon } ^ { \\bullet } \\mathbf { X }$ -split’ or $^ { 6 } \\mathrm { Z }$ -split’ rows in Table 2). The results of our baseline model on the original R2R (denoted as ‘R2R’ rows) splits are listed for comparison. As shown in Table. 2, the agent performs better on val path-seen than val path-unseen, which suggests that a path-level locality exists in current VLN agent models. Meanwhile, the results on val path-unseen are further higher than val env-unseen and it indicates the environment-level locality which is independent of the path-level locality. ",
+        "bbox": [
+            174,
+            675,
+            825,
+            786
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Existence of the Region-level Locality To further demonstrate region-level locality, we study how the success rate changes in different regions of the environment with respect to their distances to the training data, which is similar to the analysis of language ‘distance’ in Sec. 4.1. We first calculate the point-by-point shortest paths using the Dijkstra’s algorithm (Dijkstra, 1959), where the shortest distances between viewpoints $v$ and $v ^ { \\prime }$ are denoted as the graph distance $\\mathrm { d i s } _ { \\mathrm { G R A P H } } ( v , v ^ { \\prime } )$ . Based on this graph distance, we define the viewpoint distance disVIEWPOINT from a viewpoint $v$ to the training data $\\mathbb { T }$ as $v$ ’s minimal graph distance to a viewpoint $v ^ { \\prime }$ in training data. We then define the path distance $\\mathrm { d i s } _ { \\mathrm { P A T H } }$ from a validating data $x$ to the whole training data $\\mathbb { T }$ as the maximal viewpoint distance in the path of $x$ : ",
+        "bbox": [
+            174,
+            803,
+            825,
+            887
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/fb90a3896586324d3aca5aafa029005a6d8046801b735f91602e032818287107.jpg",
+        "image_caption": [
+            "Figure 4: The success rate declines as the path moves further from training regions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            287,
+            102,
+            700,
+            217
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            247,
+            823,
+            289
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e13c72dcdf9db40728af1e9266b9be1e42ea7e9eaec32062e074c0c32d969caa.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\left. { \\mathrm { d i s } _ { \\mathrm { P A T H } } ( x , \\mathbb { T } ) = \\operatorname* { m a x } _ { \\boldsymbol { v } \\in \\mathrm { p a t h } ( x ) } \\mathrm { d i s } _ { \\mathrm { V I E W P O I N T } } ( \\boldsymbol { v } , \\mathbb { T } ) } } \\\\ & { = \\operatorname* { m a x } _ { \\boldsymbol { v } \\in \\mathrm { p a t h } ( x ) } \\left\\{ \\begin{array} { l } { \\qquad \\mathrm { ~ m i n ~ } } \\\\ { \\boldsymbol { v } ^ { \\prime } \\in \\mathrm { p a t h } ( t ) } \\end{array} \\right. \\mathrm { d i s } _ { \\mathrm { G R A P H } } \\left( \\boldsymbol { v } , \\boldsymbol { v } ^ { \\prime } \\right) \\right\\} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            295,
+            681,
+            376
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We compute this path distance between paths in the env-seen validation set and training environments in our re-split data. As shown in Fig. 4, the success rate declines as the path moves further from the training environment on both re-splitting methods (i.e., $\\mathbf { \\hat { x } }$ -split’ and $^ { 6 } \\mathrm { Z }$ -split’). As conclusion, the closer the path to the training data, the higher the agent performance is, which suggests the existence of region-level locality. ",
+        "bbox": [
+            174,
+            381,
+            825,
+            452
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 WHY: WHAT INSIDE THE ENVIRONMENTS CONTRIBUTES TO THE BIAS? ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            472,
+            803,
+            488
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Sec. 4, we locate the cause of performance gap in visual environments by excluding other potential reasons and categorizing the spatial localities. However, there are still multiple possible aspects inside the environment which could lead to these spatial localities, e.g., the object layout convention and the room connections. The agent model could be biased towards the training environments by over-fitting or memorizing these environment-specific characteristics. In this section, we want to identify which aspect directly contributes to the bias and draw the following conclusion: the environment bias is attributed to low-level visual information carried by the ResNet features. ",
+        "bbox": [
+            174,
+            503,
+            825,
+            601
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We first show an experiment that effectively decreases the gap between seen and unseen environments with minimal model modifications. We then clarify our conclusions based on the findings. ",
+        "bbox": [
+            176,
+            608,
+            823,
+            636
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.1 AN INVESTIGATION EXPERIMENT: IMAGENET LABELS AS VISUAL FEATURES ",
+        "bbox": [
+            174,
+            654,
+            753,
+            667
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Suspecting that the over-fitting happens when the agent over-learns low-level features, we hope to find the replacement of ResNet 2048-features that contain minimal low-level information while preserving distinguishable visual contents. The most straightforward replacement is that instead of using mean-pooled features, we inject the frozen 1000-way classifying layer in ResNet pre-training, and use the probabilities of ImageNet labels as visual features. Shown as ‘ImageNet’ in Table. 3, the probability distribution almost closes the gap between seen and unseen. These results further constrain the reason of environment bias to the low-level ResNet features of image views. Combining with the findings of spatial localities, we suggest that environments (i.e., houses) and regions (i.e., rooms) usually have their own ‘style’. Thus the same semantic label (captured by ImageNet-1000 features) has different visual appearances (captured by ResNet features) in different environments or regions. As a result, ImageNet-1000 features, in spite of being noisy, are not distracted by low-level visual appearance and could generalize to unseen environments, while ResNet features could not. ",
+        "bbox": [
+            174,
+            680,
+            825,
+            847
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Although these ImageNet-1000 features decrease the performance gap, it has a disagreement with the VLN domain so that the validation unseen results of R4R and CVDN are slightly worse than baseline (and not much better for R2R). Hence it motivates us to find better semantic representations of environmental features that can both close the seen-unseen gap while also achieving state-of-theart on unseen results (which we discuss next). ",
+        "bbox": [
+            176,
+            854,
+            823,
+            922
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/40fd11714e83454fb41b33d752f40b626071a911bff9ece9ee0a7aeda6af925f.jpg",
+        "table_caption": [
+            "Table 3: Results showing that our semantic feature representations eliminate the performance gap in all three datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Task</td><td colspan=\"3\">Feature</td><td colspan=\"3\">Result</td></tr><tr><td>Type</td><td>Name</td><td>Dim</td><td>Val Seen</td><td>Val Unseen</td><td>Abs Gap |△|</td></tr><tr><td rowspan=\"6\">Room-to-Room</td><td>Baseline</td><td>ResNet NoDrop</td><td>2,048</td><td>54.5</td><td>38.2</td><td>16.3</td></tr><tr><td>Baseline</td><td>ResNet</td><td>2,048</td><td>56.1</td><td>47.5</td><td>8.6</td></tr><tr><td>Invesgation</td><td>ImageNet</td><td>1,000</td><td>47.1</td><td>48.2</td><td>1.1</td></tr><tr><td>Semantic</td><td>Detection</td><td>152</td><td>55.9</td><td>50.0</td><td>5.9</td></tr><tr><td>Semantic</td><td>Ground Truth</td><td>42</td><td>55.6</td><td>56.2</td><td>0.6</td></tr><tr><td>Semantic</td><td>Learned</td><td>42</td><td>53.1</td><td>53.3</td><td>0.2</td></tr><tr><td rowspan=\"6\">R4R</td><td>Baseline</td><td>ResNet NoDrop</td><td>2,048</td><td>52.5</td><td>25.8</td><td>26.7</td></tr><tr><td>Baseline</td><td>ResNet</td><td>2,048</td><td>54.6</td><td>30.7</td><td>23.9</td></tr><tr><td>Investigation</td><td>ImageNet</td><td>1,000</td><td>28.7</td><td>28.9</td><td>0.2</td></tr><tr><td>Semantic</td><td>Detection</td><td>152</td><td>48.8</td><td>32.0</td><td>16.8</td></tr><tr><td>Semantic</td><td>Ground Truth</td><td>42</td><td>47.6</td><td>35.9</td><td>11.7</td></tr><tr><td>Semantic</td><td>Learned</td><td>42</td><td>36.2</td><td>36.1</td><td>0.1</td></tr><tr><td rowspan=\"6\">CVDN</td><td>Baseline</td><td>ResNet NoDrop</td><td>2,048</td><td>5.88</td><td>2.14</td><td>3.74</td></tr><tr><td>Baseline</td><td>ResNet</td><td>2,048</td><td>5.97</td><td>2.23</td><td>3.74</td></tr><tr><td>Invesgation</td><td>ImageNet</td><td>1,000</td><td>3.22</td><td>2.08</td><td>1.14</td></tr><tr><td>Semantic</td><td>Detection</td><td>152</td><td>3.34</td><td>2.08</td><td>1.26</td></tr><tr><td>Semantic</td><td>Ground Truth</td><td>42</td><td>3.75</td><td>2.69</td><td>1.06</td></tr><tr><td>Semantic</td><td>Learned</td><td>42</td><td>2.60</td><td>2.43</td><td>0.17</td></tr></table>",
+        "bbox": [
+            176,
+            137,
+            820,
+            420
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 HOW: METHODOLOGY TO FIX THE ENVIRONMENT BIAS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            435,
+            676,
+            452
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In the previous section (Sec. 5), we found that the environment bias is related to the low-level visual features (i.e., 2048-dim ResNet features). Following the findings we observed in Sec. 5.1, we build our agent on the features which are more correlated to the VLN environmental semantics than the ImageNet label features in Sec. 5.1. We first demonstrate our baseline results on three VLN datasets and then explore the advanced semantic feature replacements. As shown in Table 3, these advanced semantic features could effectively reduce the performance gap between seen and unseen environments and improve the unseen results compared to our strong baselines. The effectiveness of these semantic features supports our explanation of the environment bias in Sec. 5 and also suggests that future work in VLN tasks should think about such generalization issues. ",
+        "bbox": [
+            173,
+            467,
+            825,
+            592
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.1 BASELINE",
+        "text_level": 1,
+        "bbox": [
+            174,
+            609,
+            285,
+            623
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In our baseline model, following the previous works we use the standard ResNet features as the representation of environments (Anderson et al., 2018b; Jain et al., 2019; Thomason et al., 2019b). These features come from the mean-pooled layer after the final convolutional layer of ResNet152 (He et al., 2016) pre-trained on ImageNet (Russakovsky et al., 2015). As shown in ‘Baseline’5 rows of Table. 3, val-seen results are significantly higher than val-unseen results in all three datasets. Note that our baseline method takes the ‘feature dropout’ technique demonstrated in Tan et al. (2019) (without back translation): the ResNet features are randomly masked by zero before used as inputs of the agent. Without this ‘feature dropout’ (denoted as ‘ResNet NoDrop’ in Table. 3), the gaps will increase in R2R and R4R, which suggests that this ‘feature dropout’ technique also helps to eliminate the low-level visual information over-fitting as we discussed in Sec. 5. However, the performance gap is still large, which leads us to the following discussions of semantic features. ",
+        "bbox": [
+            174,
+            636,
+            825,
+            787
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.2 DETECTED OBJECTS AREAS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            805,
+            411,
+            819
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "During navigation, the objects in the environments are crucial since their matchings with the instruction often indicate the locations that can guide the agent, thus object detection results of the environments can provide relevant semantic information. In our work, we utilize the detection information generated by Faster R-CNN (Ren et al., 2015) to create the feature representations. Comparing to ",
+        "bbox": [
+            174,
+            832,
+            823,
+            887
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ImageNet-1000 features (Sec. 5.1), these detection features include more environmental information since the viewing images in VLN usually contain multiple objects. Instead of directly using classification probabilities of the labels from ResNet and different from the approach in $\\mathrm { H u }$ et al. (2019) who utilized the embeddings of detected labels, we design our detection features f DETECT of each image view as the sum of the areas of detected objects weighted by detection confidence: ",
+        "bbox": [
+            173,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/64c0f457e217b81a5a46ad64844187f77fdb06477e6b0fc0c34fd70e9dffcc33.jpg",
+        "text": "$$\n\\mathrm { f _ { \\mathrm { { D E T E C T } } } } \\mathrm { = } [ a _ { c _ { 1 } } , a _ { c _ { 2 } } , \\dotsc , a _ { c _ { n } } ] ; \\qquad a _ { c _ { i } } = \\sum _ { \\mathrm { o b j i s } ~ c _ { i } } \\mathrm { { A r e a } ( o b j ) } \\cdot \\mathrm { { C o n f ( o b j ) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
+            176,
+            717,
+            212
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where the $c _ { i }$ and $\\boldsymbol { a } _ { c _ { i } }$ are the label and feature of each detected object, $\\mathrm { A r e a } ( * )$ and $\\operatorname { C o n f } ( * )$ are the area and confidence of each object. For implementation details, we use the Faster R-CNN (Ren et al., 2015) trained on Visual Genome (Krishna et al., 2017) provided in Bottom-Up Attention (Anderson et al., 2018a). To eliminate the labels irrelevant to VLN task, we calculate the total areas of each detection object among all environments and pick the labels that take up a relatively large proportion of the environments, creating features of dimension 152.6 Denoted as ‘Detection’ in Table 3, the performance gap is diminished with these detection features compared to baselines in all three datasets, indicating that changing the features to a higher semantic level has a positive effect on alleviating the environment bias. Meanwhile, the improvement of unseen validation results on R2R an R4R datasets suggests the better efficiency in the VLN task than the ImageNet labels. ",
+        "bbox": [
+            173,
+            215,
+            825,
+            354
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.3 SEMANTIC SEGMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            371,
+            403,
+            385
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Although the detection features can provide adequate semantic information for the agent to achieve comparable results as the baseline model, they do not fully utilize the visual information where the content left over from detection may contain useful knowledge for navigation. A better semantic representation is the semantic segmentation, which segments each view image on the pixel level and gives the label to each segment region, allowing us to utilize the semantics from the entire environment. Matterport3D (Chang et al., 2017) dataset provides the labeled semantic segmentation information of every scene and we take the rendered images from Tan et al. (2019)7. A comparison example of RGB images and semantic views is available in the Appendix. Since the semantic segmentation images are fine-grained and blurry in boundaries, we follow the design of detection features, using the areas of semantic classes in each image view as the semantic features (confidence is excluded since semantic segmentation does not provide this value). The areas are normalized to $[ 0 , 1 ]$ by dividing the area of the whole image region. We first assume that the semantic information is provided as additional environmental information and the results of the model using the ground truth semantic areas are shown in the ‘ground truth’ rows in Table. 3. We next study the situation where the semantic information is not available in testing environments thus the information needs to be learned from training environments. Thus we train a separate multi-layer perceptron to predict the areas of these semantic classes (details in Appendix), and the results of the model with these predicted semantics as features are shown in ‘learned’. As shown in Table. 3, both ‘ground truth’ and ‘learned’ semantic representations bring the performance of seen and unseen closer comparing to the baseline model, and the smallest performance gaps come from learned semantic segmentation features in all three datasets. The highest validation unseen success rates among all the proposed feature representations are also produced by semantic segmentation features, ‘learned’ semantic for R4R and ‘ground truth’ semantic for R2R and CVDN. Overall, among all the semantic representations we have explored, the semantic segmentation features are most effective in eliminating the environment bias. ",
+        "bbox": [
+            173,
+            397,
+            825,
+            742
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            762,
+            318,
+            779
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we focus on studying the performance gap between seen and unseen environments widely observed in vision-and-language navigation (VLN) tasks, trying to find where and why this environment bias exists and provide possible initial solutions. By designing the diagnosis experiments of environment re-splitting and feature replacement, we locate the environment bias to be in the low-level visual appearance; and we discuss semantic features that decrease the performance gap in three VLN datasets and achieve state-of-the-art results. ",
+        "bbox": [
+            174,
+            794,
+            825,
+            877
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            287,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Peter Anderson, Xiaodong He, Chris Buehler, Damien Teney, Mark Johnson, Stephen Gould, and Lei Zhang. Bottom-up and top-down attention for image captioning and visual question answering. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 6077–6086, 2018a. ",
+        "bbox": [
+            173,
+            126,
+            825,
+            181
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Peter Anderson, Qi Wu, Damien Teney, Jake Bruce, Mark Johnson, Niko Sunderhauf, Ian Reid, ¨ Stephen Gould, and Anton van den Hengel. Vision-and-language navigation: Interpreting visually-grounded navigation instructions in real environments. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3674–3683, 2018b. ",
+        "bbox": [
+            173,
+            193,
+            825,
+            250
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Peter Anderson, Ayush Shrivastava, Devi Parikh, Dhruv Batra, and Stefan Lee. Chasing ghosts: Instruction following as bayesian state tracking. In Advances in Neural Information Processing Systems (NeurIPS), 2019. ",
+        "bbox": [
+            173,
+            260,
+            825,
+            303
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In The International Conference on Learning Representations (ICLR), 2015. ",
+        "bbox": [
+            173,
+            313,
+            825,
+            354
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Gilles Blanchard, Gyemin Lee, and Clayton Scott. Generalizing from several related classification tasks to a new unlabeled sample. In Advances in neural information processing systems, pp. 2178–2186, 2011. ",
+        "bbox": [
+            173,
+            364,
+            823,
+            409
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Gilles Blanchard, Aniket Anand Deshmukh, Urun Dogan, Gyemin Lee, and Clayton Scott. Domain generalization by marginal transfer learning. arXiv preprint arXiv:1711.07910, 2017. ",
+        "bbox": [
+            171,
+            417,
+            823,
+            448
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Valts Blukis, Dipendra Misra, Ross A Knepper, and Yoav Artzi. Mapping navigation instructions to continuous control actions with position-visitation prediction. In Conference on Robot Learning, pp. 505–518, 2018. ",
+        "bbox": [
+            173,
+            458,
+            825,
+            500
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Fabio M Carlucci, Antonio D’Innocente, Silvia Bucci, Barbara Caputo, and Tatiana Tommasi. Domain generalization by solving jigsaw puzzles. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2229–2238, 2019. ",
+        "bbox": [
+            173,
+            510,
+            825,
+            554
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Angel Chang, Angela Dai, Thomas Funkhouser, Maciej Halber, Matthias Niebner, Manolis Savva, Shuran Song, Andy Zeng, and Yinda Zhang. Matterport3d: Learning from rgb-d data in indoor environments. In 2017 International Conference on 3D Vision (3DV), pp. 667–676. IEEE, 2017. ",
+        "bbox": [
+            173,
+            563,
+            823,
+            607
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chao Chen, Zhihong Chen, Boyuan Jiang, and Xinyu Jin. Joint domain alignment and discriminative feature learning for unsupervised deep domain adaptation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 3296–3303, 2019a. ",
+        "bbox": [
+            176,
+            616,
+            823,
+            660
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chaoqi Chen, Weiping Xie, Wenbing Huang, Yu Rong, Xinghao Ding, Yue Huang, Tingyang Xu, and Junzhou Huang. Progressive feature alignment for unsupervised domain adaptation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 627–636, 2019b. ",
+        "bbox": [
+            173,
+            669,
+            825,
+            726
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Howard Chen, Alane Suhr, Dipendra Misra, Noah Snavely, and Yoav Artzi. Touchdown: Natural language navigation and spatial reasoning in visual street environments. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 12538–12547, 2019c. ",
+        "bbox": [
+            173,
+            736,
+            825,
+            780
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Abhishek Das, Samyak Datta, Georgia Gkioxari, Stefan Lee, Devi Parikh, and Dhruv Batra. Embodied question answering. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp. 2054–2063, 2018. ",
+        "bbox": [
+            173,
+            789,
+            823,
+            832
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Aniket Anand Deshmukh, Yunwen Lei, Srinagesh Sharma, Urun Dogan, James W Cutler, and Clayton Scott. A generalization error bound for multi-class domain generalization. arXiv preprint arXiv:1905.10392, 2019. ",
+        "bbox": [
+            173,
+            842,
+            823,
+            885
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Edsger W Dijkstra. A note on two problems in connexion with graphs. Numerische mathematik, 1 (1):269–271, 1959. ",
+        "bbox": [
+            171,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Daniel Fried, Ronghang Hu, Volkan Cirik, Anna Rohrbach, Jacob Andreas, Louis-Philippe Morency, Taylor Berg-Kirkpatrick, Kate Saenko, Dan Klein, and Trevor Darrell. Speaker-follower models for vision-and-language navigation. In Advances in Neural Information Processing Systems, pp. 3314–3325, 2018. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            160
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Rui Gong, Wen Li, Yuhua Chen, and Luc Van Gool. Dlow: Domain flow for adaptation and generalization. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2477–2486, 2019. ",
+        "bbox": [
+            171,
+            170,
+            823,
+            213
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014. ",
+        "bbox": [
+            171,
+            224,
+            823,
+            267
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Daniel Gordon, Aniruddha Kembhavi, Mohammad Rastegari, Joseph Redmon, Dieter Fox, and Ali Farhadi. Iqa: Visual question answering in interactive environments. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4089–4098, 2018. ",
+        "bbox": [
+            178,
+            277,
+            823,
+            321
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016. ",
+        "bbox": [
+            174,
+            330,
+            823,
+            375
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ehsan Hosseini-Asl, Yingbo Zhou, Caiming Xiong, and Richard Socher. Augmented cyclic adversarial learning for low resource domain adaptation. In The International Conference on Learning Representations (ICLR), 2019. ",
+        "bbox": [
+            173,
+            385,
+            825,
+            428
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ronghang Hu, Daniel Fried, Anna Rohrbach, Dan Klein, Kate Saenko, et al. Are you looking? grounding to multiple modalities in vision-and-language navigation. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, 2019. ",
+        "bbox": [
+            173,
+            438,
+            825,
+            482
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Haoshuo Huang, Vihan Jain, Harsh Mehta, Alexander Ku, Gabriel Magalhaes, Jason Baldridge, and Eugene Ie. Transferable representation learning in vision-and-language navigation. In IEEE International Conference on Computer Vision, 2019. ",
+        "bbox": [
+            174,
+            492,
+            823,
+            535
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Vihan Jain, Gabriel Magalhaes, Alexander Ku, Ashish Vaswani, Eugene Ie, and Jason Baldridge. Stay on the path: Instruction fidelity in vision-and-language navigation. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), 2019. ",
+        "bbox": [
+            174,
+            545,
+            825,
+            602
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Liyiming Ke, Xiujun Li, Yonatan Bisk, Ari Holtzman, Zhe Gan, Jingjing Liu, Jianfeng Gao, Yejin Choi, and Siddhartha Srinivasa. Tactical rewind: Self-correction via backtracking in vision-andlanguage navigation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 6741–6749, 2019. ",
+        "bbox": [
+            174,
+            613,
+            825,
+            670
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ranjay Krishna, Yuke Zhu, Oliver Groth, Justin Johnson, Kenji Hata, Joshua Kravitz, Stephanie Chen, Yannis Kalantidis, Li-Jia Li, David A Shamma, et al. Visual genome: Connecting language and vision using crowdsourced dense image annotations. International Journal of Computer Vision, 123(1):32–73, 2017. ",
+        "bbox": [
+            174,
+            680,
+            825,
+            737
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M Hospedales. Deeper, broader and artier domain generalization. In Proceedings of the IEEE International Conference on Computer Vision, pp. 5542–5550, 2017. ",
+        "bbox": [
+            173,
+            747,
+            826,
+            791
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M Hospedales. Learning to generalize: Metalearning for domain generalization. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018. ",
+        "bbox": [
+            173,
+            801,
+            821,
+            844
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Chin-Yew Lin. Rouge: A package for automatic evaluation of summaries. In Text summarization branches out, pp. 74–81, 2004. ",
+        "bbox": [
+            171,
+            854,
+            823,
+            885
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Mingsheng Long, Zhangjie Cao, Jianmin Wang, and Michael I Jordan. Conditional adversarial domain adaptation. In Advances in Neural Information Processing Systems, pp. 1640–1650, 2018. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Chih-Yao Ma, Jiasen Lu, Zuxuan Wu, Ghassan AlRegib, Zsolt Kira, Richard Socher, and Caiming Xiong. Self-monitoring navigation agent via auxiliary progress estimation. In The International Conference on Learning Representations (ICLR), 2019a. ",
+        "bbox": [
+            178,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Chih-Yao Ma, Zuxuan Wu, Ghassan AlRegib, Caiming Xiong, and Zsolt Kira. The regretful agent: Heuristic-aided navigation through progress estimation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 6732–6740, 2019b. ",
+        "bbox": [
+            173,
+            156,
+            825,
+            200
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Matt MacMahon, Brian Stankiewicz, and Benjamin Kuipers. Walk the talk: connecting language, knowledge, and action in route instructions. In proceedings of the 21st national conference on Artificial intelligence-Volume 2, pp. 1475–1482. AAAI Press, 2006. ",
+        "bbox": [
+            173,
+            210,
+            825,
+            253
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Piotr Mirowski, Matt Grimes, Mateusz Malinowski, Karl Moritz Hermann, Keith Anderson, Denis Teplyashin, Karen Simonyan, Andrew Zisserman, Raia Hadsell, et al. Learning to navigate in cities without a map. In Advances in Neural Information Processing Systems, pp. 2419–2430, 2018. ",
+        "bbox": [
+            173,
+            263,
+            825,
+            320
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Dipendra Misra, Andrew Bennett, Valts Blukis, Eyvind Niklasson, Max Shatkhin, and Yoav Artzi. Mapping instructions to actions in 3d environments with visual goal prediction. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pp. 2667–2678, 2018. ",
+        "bbox": [
+            174,
+            332,
+            825,
+            388
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Krikamol Muandet, David Balduzzi, and Bernhard Scholkopf. Domain generalization via invariant ¨ feature representation. In International Conference on Machine Learning, pp. 10–18, 2013. ",
+        "bbox": [
+            173,
+            398,
+            825,
+            429
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. Bleu: a method for automatic evaluation of machine translation. In Proceedings of the 40th annual meeting on association for computational linguistics, pp. 311–318. Association for Computational Linguistics, 2002. ",
+        "bbox": [
+            174,
+            439,
+            823,
+            482
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Yuankai Qi, Qi Wu, Peter Anderson, Marco Liu, Chunhua Shen, and Anton van den Hengel. Rerere: Remote embodied referring expressions in real indoor environments. arXiv preprint arXiv:1904.10151, 2019. ",
+        "bbox": [
+            174,
+            493,
+            825,
+            535
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in neural information processing systems, pp. 91–99, 2015. ",
+        "bbox": [
+            174,
+            546,
+            825,
+            589
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Artem Rozantsev, Mathieu Salzmann, and Pascal Fua. Residual parameter transfer for deep domain adaptation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4339–4348, 2018. ",
+        "bbox": [
+            174,
+            601,
+            825,
+            643
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211–252, 2015. ",
+        "bbox": [
+            176,
+            654,
+            823,
+            698
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Hao Tan, Licheng Yu, and Mohit Bansal. Learning to navigate unseen environments: Back translation with environmental dropout. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pp. 2610–2621, 2019. ",
+        "bbox": [
+            173,
+            707,
+            825,
+            765
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jesse Thomason, Daniel Gordan, and Yonatan Bisk. Shifting the baseline: Single modality performance on visual navigation & qa. In NACCL, 2019a. ",
+        "bbox": [
+            173,
+            775,
+            823,
+            804
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jesse Thomason, Michael Murray, Maya Cakmak, and Luke Zettlemoyer. Vision-and-dialog navigation. In CoRL, 2019b. ",
+        "bbox": [
+            171,
+            815,
+            823,
+            844
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Daixin Wang, Peng Cui, and Wenwu Zhu. Deep asymmetric transfer network for unbalanced domain adaptation. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018a. ",
+        "bbox": [
+            171,
+            854,
+            823,
+            885
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Ximei Wang, Liang Li, Weirui Ye, Mingsheng Long, and Jianmin Wang. Transferable attention for domain adaptation. In AAAI Conference on Artificial Intelligence (AAAI), 2019a. ",
+        "bbox": [
+            173,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Xin Wang, Wenhan Xiong, Hongmin Wang, and William Yang Wang. Look before you leap: Bridging model-free and model-based reinforcement learning for planned-ahead vision-and-language navigation. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 37–53, 2018b. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            159
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Xin Wang, Qiuyuan Huang, Asli Celikyilmaz, Jianfeng Gao, Dinghan Shen, Yuan-Fang Wang, William Yang Wang, and Lei Zhang. Reinforced cross-modal matching and self-supervised imitation learning for vision-language navigation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 6629–6638, 2019b. ",
+        "bbox": [
+            173,
+            167,
+            825,
+            224
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Yi Wu, Yuxin Wu, Georgia Gkioxari, and Yuandong Tian. Building generalizable agents with a realistic and rich 3d environment. arXiv preprint arXiv:1801.02209, 2018. ",
+        "bbox": [
+            174,
+            232,
+            823,
+            261
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Licheng Yu, Xinlei Chen, Georgia Gkioxari, Mohit Bansal, Tamara L Berg, and Dhruv Batra. Multitarget embodied question answering. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 6309–6318, 2019. ",
+        "bbox": [
+            173,
+            268,
+            823,
+            311
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Yabin Zhang, Hui Tang, Kui Jia, and Mingkui Tan. Domain-symmetric networks for adversarial domain adaptation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5031–5040, 2019. ",
+        "bbox": [
+            174,
+            319,
+            825,
+            361
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pp. 2223–2232, 2017. ",
+        "bbox": [
+            174,
+            369,
+            825,
+            412
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            438,
+            297,
+            452
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/aeffcbb03294a1d8c8132f9852d689ea8d2bd272e7c64a054ba954b51b1c77a8.jpg",
+        "image_caption": [
+            "Figure 5: Comparisons between RGB images and their semantic views. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            254,
+            474,
+            743,
+            618
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.1 EXAMPLES OF RGB IMAGES AND SEMANTIC VIEWS ",
+        "bbox": [
+            174,
+            670,
+            583,
+            684
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 5, we show a rendered semantic view from Tan et al. (2019) and its original RGB image. Different colors indicate different semantic segmentation areas and 40 semantic labels are considered in the Matterport3D dataset Chang et al. (2017). ",
+        "bbox": [
+            174,
+            695,
+            823,
+            738
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2 DETAILS OF ‘LEARNED’ SEMANTIC TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            755,
+            542,
+            768
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We use a multi-layer perceptron over the ResNet features to generate the ‘learned’ semantic features. The multi-layer perceptron includes three fully-connected layers with ReLU activation on the outputs of the first two layers. The input is the 2048-dim ResNet feature $f$ of each image view. The hidden sizes of the first two layers are 512 and 128. The final layer will output the 42-dim semantic feature $y$ that represents the areas of each semantic class. After the linear layers, we use the sigmoid function $\\sigma$ to convert the output to the ratio of areas. ",
+        "bbox": [
+            174,
+            780,
+            825,
+            864
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4f213a2bccc227656c70eabc0c5768fae29ada90e2373849a74ad6645cd88cf9.jpg",
+        "text": "$$\n\\begin{array} { c } { { x _ { 1 } = \\mathrm { R e L U } ( A _ { 1 } f + b _ { 1 } ) } } \\\\ { { x _ { 2 } = \\mathrm { R e L U } ( A _ { 2 } x _ { 1 } + b _ { 2 } ) } } \\\\ { { y = \\sigma ( A _ { 3 } x _ { 2 } + b _ { 3 } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
+            867,
+            583,
+            922
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The model is trained with ground truth semantic areas $y _ { \\mathrm { A R E A } }$ (normalized to $[ 0 , 1 ] )$ and only the views in training environments are used in training. We minimize the binary cross-entropy loss between the ground truth areas $\\{ y _ { i } ^ { * } \\}$ and the predicted areas $\\{ y _ { i } \\}$ , where $i$ indicate the $i$ -th semantic class. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            147
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3dd4cba5ff98958bd14fc9065b8e4c8b10edb254d329be878435515daf6e1d5f.jpg",
+        "text": "$$\n\\mathcal { L } = - \\sum _ { i } \\left( y _ { i } ^ { * } \\log y _ { i } + \\left( 1 - y _ { i } ^ { * } \\right) \\log \\left( 1 - y _ { i } \\right) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            152,
+            650,
+            186
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Dropout layers with a probability of 0.5 are added between fully-connected layers while training.   \nThe sigmoid function $\\sigma$ and the cross-entropy loss are combined to improve numerical stability. ",
+        "bbox": [
+            171,
+            193,
+            821,
+            222
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "After the model is fitted, we freeze the weight and use it to predict the semantic features of all seen and unseen environments (i.e., environments for training, val-seen, and val-unseen data). The predicted features are then used as the input of our neural agent model for different datasets (i.e., R2R, R4R, and CVDN), and the neural agent models are the same except we change the input dimension from 2048 (the dimension of ResNet features) to 42 (the number of semantic classes). ",
+        "bbox": [
+            174,
+            227,
+            825,
+            297
+        ],
+        "page_idx": 14
+    }
+]

parse/train/S1evHerYPr/S1evHerYPr.md

@@ -0,0 +1,421 @@
+# IMPROVING GENERALIZATION IN META REINFORCEMENT LEARNING USING LEARNED OBJECTIVES
+
+Louis Kirsch, Sjoerd van Steenkiste, Jurgen Schmidhuber ¨ The Swiss AI Lab IDSIA, USI, SUPSI {louis, sjoerd, juergen}@idsia.ch
+
+# ABSTRACT
+
+Biological evolution has distilled the experiences of many learners into the general learning algorithms of humans. Our novel meta reinforcement learning algorithm MetaGenRL is inspired by this process. MetaGenRL distills the experiences of many complex agents to meta-learn a low-complexity neural objective function that decides how future individuals will learn. Unlike recent meta-RL algorithms, MetaGenRL can generalize to new environments that are entirely different from those used for meta-training. In some cases, it even outperforms humanengineered RL algorithms. MetaGenRL uses off-policy second-order gradients during meta-training that greatly increase its sample efficiency.
+
+# 1 INTRODUCTION
+
+The process of evolution has equipped humans with incredibly general learning algorithms. They enable us to solve a wide range of problems, even in the absence of a large number of related prior experiences. The algorithms that give rise to these capabilities are the result of distilling the collective experiences of many learners throughout the course of natural evolution. By essentially learning from learning experiences in this way, the resulting knowledge can be compactly encoded in the genetic code of an individual to give rise to the general learning capabilities that we observe today.
+
+In contrast, Reinforcement Learning (RL) in artificial agents rarely proceeds in this way. The learning rules that are used to train agents are the result of years of human engineering and design, (e.g. Williams (1992); Wierstra et al. (2008); Mnih et al. (2013); Lillicrap et al. (2016); Schulman et al. (2015a)). Correspondingly, artificial agents are inherently limited by the ability of the designer to incorporate the right inductive biases in order to learn from previous experiences.
+
+Several works have proposed an alternative framework based on meta reinforcement learning (Schmidhuber, 1994; Wang et al., 2016; Duan et al., 2016; Finn et al., 2017; Houthooft et al., 2018; Clune, 2019). Meta-RL distinguishes between learning to act in the environment (the reinforcement learning problem) and learning to learn (the meta-learning problem). Hence, learning itself is now a learning problem, which in principle allows one to leverage prior learning experiences to meta-learn general learning rules that surpass human-engineered alternatives. However, while prior work found that learning rules could be meta-learned that generalize to slightly different environments or goals (Finn et al., 2017; Plappert et al., 2018; Houthooft et al., 2018), generalization to entirely different environments remains an open problem.
+
+In this paper we present MetaGenRL1, a novel meta reinforcement learning algorithm that metalearns learning rules that generalize to entirely different environments. MetaGenRL is inspired by the process of natural evolution as it distills the experiences of many agents into the parameters of an objective function that decides how future individuals will learn. Similar to Evolved Policy Gradients (EPG; Houthooft et al. (2018)), it meta-learns low complexity neural objective functions that can be used to train complex agents with many parameters. However, unlike EPG, it is able to meta-learn using second-order gradients, which offers several advantages as we will demonstrate.
+
+We evaluate MetaGenRL on a variety of continuous control tasks and compare to $\mathrm { { R L ^ { 2 } } }$ (Wang et al., 2016; Duan et al., 2016) and EPG in addition to several human engineered learning algorithms.
+
+Compared to $\mathtt { R L } ^ { 2 }$ we find that MetaGenRL does not overfit and is able to train randomly initialized agents using meta-learned learning rules on entirely different environments. Compared to EPG we find that MetaGenRL is more sample efficient, and outperforms significantly under a fixed budget of environment interactions. The results of an ablation study and additional analysis provide further insight into the benefits of our approach.
+
+# 2 PRELIMINARIES
+
+Notation We consider the standard MDP Reinforcement Learning setting defined by a tuple $e =$ $( S , A , P , \rho _ { 0 } , r , \gamma , T )$ consisting of states $S$ , actions $A$ , the transition probability distribution $P :$ $S \times A \times S \to \mathbb { R } _ { + } .$ , an initial state distribution $\rho _ { 0 } : S \to \mathbb { R } _ { + }$ , the reward function $r : S \times A \to$ $[ - R _ { m a x } , R _ { m a x } ]$ , a discount factor $\gamma$ , and the episode length $T$ . The objective for the probabilistic policy $\pi _ { \phi } : S \times A \to \mathbb { R } _ { + }$ parameterized by $\phi$ is to maximize the expected discounted return:
+
+$\mathbb { E } _ { \tau } [ \sum _ { t = 0 } ^ { T - 1 } \gamma ^ { t } r _ { t } ]$ $[ \sum \gamma ^ { t } r _ { t } ] , \mathrm { ~ w h e r e ~ } s _ { 0 } \sim \rho _ { 0 } ( s _ { 0 } ) , a _ { t } \sim \pi _ { \phi } ( a _ { t } | s _ { t } ) , s _ { t + 1 } \sim P ( s _ { t + 1 } | s _ { t } , a _ { t } ) , r _ { t } = r ( s _ { t } , a _ { t } ) ,$ with $\tau = ( s _ { 0 } , a _ { 0 } , r _ { 0 } , s _ { 1 } , . . . , s _ { T - 1 } , a _ { T - 1 } , r _ { T - 1 } ) .$
+
+Human Engineered Gradient Estimators A popular gradient-based approach to maximizing Equation 1 is REINFORCE (Williams, 1992). It directly differentiates Equation 1 with respect to $\phi$ using the likelihood ratio trick to derive gradient estimates of the form:
+
+$$
+\nabla _ { \phi } \mathbb { E } _ { \tau } \big [ L _ { R E I N F } ( \tau , \pi _ { \phi } ) \big ] : = \mathbb { E } _ { \tau } \big [ \nabla _ { \phi } \sum _ { t = 0 } ^ { T - 1 } \log \pi _ { \phi } ( a _ { t } | s _ { t } ) \cdot \sum _ { t ^ { \prime } = t } ^ { T - 1 } \gamma ^ { t ^ { \prime } - t } r _ { t ^ { \prime } } ) \big ] .
+$$
+
+Although this basic estimator is rarely used in practice, it has become a building block for an entire class of policy-gradient algorithms of this form. For example, a popular extension from Schulman et al. (2015b) combines REINFORCE with a Generalized Advantage Estimate (GAE) to yield the following policy gradient estimator:
+
+$$
+\nabla _ { \phi } \mathbb { E } _ { \tau } [ L _ { G A E } ( \tau , \pi _ { \phi } , V ) ] : = \mathbb { E } _ { \tau } [ \nabla _ { \phi } \sum _ { t = 0 } ^ { T - 1 } \log \pi _ { \phi } ( a _ { t } \vert s _ { t } ) \cdot A ( \tau , V , t ) ] .
+$$
+
+where $A ( \tau , V , t )$ is the GAE and $V : S  \mathbb { R }$ is a value function estimate. Several recent other extensions include TRPO (Schulman et al., 2015a), which discourages bad policy updates using trust regions and iterative off-policy updates, or PPO (Schulman et al., 2017), which offers similar benefits using only first order approximations.
+
+Parametrized Objective Functions In this work we note that many of these human engineered policy gradient estimators can be viewed as specific implementations of a general objective function $L$ that is differentiated with respect to the policy parameters:
+
+$$
+\begin{array} { r } { \nabla _ { \phi } \mathbb { E } _ { \tau } [ L ( \tau , \pi _ { \phi } , V ) ] . } \end{array}
+$$
+
+Hence, it becomes natural to consider a generic parametrization of $L$ that, for various choices of parameters $\alpha$ , recovers some of these estimators. In this paper, we will consider neural objective functions where $L _ { \alpha }$ is implemented by a neural network. Our goal is then to optimize the parameters $\alpha$ of this neural network in order to give rise to a new learning algorithm that best maximizes Equation 1 on an entire class of (different) environments.
+
+# 3 META-LEARNING NEURAL OBJECTIVES
+
+In this work we propose MetaGenRL, a novel meta reinforcement learning algorithm that metalearns neural objective functions of the form $L _ { \alpha } ( \tau , \pi _ { \phi } , V )$ . MetaGenRL makes use of value functions and second-order gradients, which makes it more sample efficient compared to prior work (Duan et al., 2016; Wang et al., 2016; Houthooft et al., 2018). More so, as we will demonstrate, MetaGenRL meta-learns objective functions that generalize to vastly different environments.
+
+![](images/61813d5bb79360ec83c55d7c4e6a68012de892729ebe4fadacfcd4e4d5846714.jpg)  
+Figure 1: A schematic of MetaGenRL. On the left a population of agents $( i \in { 1 , \ldots , N } )$ , where each member consist of a critic ${ Q } _ { \theta } ^ { ( i ) }$ and a policy $\pi _ { \phi } ^ { ( i ) }$ that interact with a particular environment $e ^ { ( i ) }$ and store collected data in a corresponding replay buffer $B ^ { ( i ) }$ . On the right a meta-learned neural objective function $L _ { \alpha }$ that is shared across the population. Learning (dotted arrows) proceeds as follows: Each policy is updated by differentiating $L _ { \alpha }$ , while the critic is updated using the usual TD-error (not shown). $L _ { \alpha }$ is meta-learned by computing second-order gradients that can be obtained by differentiating through the critic.
+
+Our key insight is that a differentiable critic $Q _ { \theta } : S \times A  \mathbb { R }$ can be used to measure the effect of locally changing the objective function parameters $\alpha$ based on the quality of the corresponding policy gradients. This enables a population of agents to use and improve a single parameterized objective function $L _ { \alpha }$ through interacting with a set of (potentially different) environments. During evaluation (meta-test time), the meta-learned objective function can then be used to train a randomly initialized RL agent in a new environment.
+
+# 3.1 FROM DDPG TO GRADIENT-BASED META-LEARNING OF NEURAL OBJECTIVES
+
+We will formally introduce MetaGenRL as an extension of the DDPG actor-critic framework (Silver et al., 2014; Lillicrap et al., 2016). In DDPG, a parameterized critic of the form $Q _ { \theta } : S \times A  \mathbb { R }$ transforms the non-differentiable RL reward maximization problem into a myopic value maximization problem for any $s _ { t } \in S$ . This is done by alternating between optimization of the critic $Q _ { \theta }$ and the (here deterministic) policy $\pi _ { \phi }$ . The critic is trained to minimize the TD-error by following:
+
+$$
+\nabla _ { \theta } \sum _ { ( s _ { t } , a _ { t } , r _ { t } , s _ { t + 1 } ) } ( Q _ { \theta } ( s _ { t } , a _ { t } ) - y _ { t } ) ^ { 2 } , \mathrm { w h e r e } y _ { t } = r _ { t } + \gamma \cdot Q _ { \theta } \big ( s _ { t + 1 } , \pi _ { \phi } \big ( s _ { t + 1 } \big ) \big ) ,
+$$
+
+and the dependence of $y _ { t }$ on the parameter vector $\theta$ is ignored. The policy $\pi _ { \phi }$ is improved to increase the expected return from arbitrary states by following the gradient $\begin{array} { r } { \nabla _ { \phi } \sum _ { s _ { t } } Q _ { \theta } \big ( s _ { t } , \pi _ { \phi } ( s _ { t } ) \big ) } \end{array}$ . Both gradients can be computed entirely off-policy by sampling trajectories from a replay buffer.
+
+MetaGenRL builds on this idea of differentiating the critic $Q _ { \theta }$ with respect to the policy parameters. It incorporates a parameterized objective function $L _ { \alpha }$ that is used to improve the policy (i.e. by following the gradient $\nabla _ { \phi } L _ { \alpha } )$ , which adds one extra level of indirection: The critic $Q _ { \theta }$ improves $L _ { \alpha }$ , while $L _ { \alpha }$ improves the policy $\pi _ { \phi }$ . By first differentiating with respect to the objective function parameters $\alpha$ , and then with respect to the policy parameters $\phi$ , the critic can be used to measure the effect of updating $\pi _ { \phi }$ using $L _ { \alpha }$ on the estimated return2:
+
+$$
+\nabla _ { \alpha } Q _ { \theta } \bigl ( s _ { t } , \pi _ { \phi ^ { \prime } } \bigl ( s _ { t } \bigr ) \bigr ) , \mathrm { w h e r e } \phi ^ { \prime } = \phi - \nabla _ { \phi } L _ { \alpha } ( \tau , x ( \phi ) , V ) .
+$$
+
+This constitutes a type of second order gradient $\nabla _ { \alpha } \nabla _ { \phi }$ that can be used to meta-train $L _ { \alpha }$ to provide better updates to the policy parameters in the future. In practice we will use batching to optimize Equation 6 over multiple trajectories $\tau$ .
+
+Similarly to the policy-gradient estimators from Section 2, the objective function $L _ { \alpha } ( \tau , x ( \phi ) , V )$ receives as inputs an episode trajectory $\tau = ( s _ { 0 : T - 1 } , a _ { 0 : T - 1 } , r _ { 0 : T - 1 } )$ , the value function estimates
+
+<table><tr><td colspan="2">Algorithm1MetaGenRL:Meta-Training</td></tr><tr><td>Require: p(e) a distribution of environments P←{(ei~p(e),Φ1,01,B1←O),..} Randomly initialize objective function Lα while L has not converged do</td><td>&gt;Randomly initialize population of agents</td></tr><tr><td>for e,Φ,0,B∈Pdo if extend replay bufferB then</td><td>For each agent iin parallel</td></tr><tr><td>Extend Busing T in e</td><td></td></tr><tr><td>Sample trajectories from B Update critic Qe using TD-error</td><td></td></tr><tr><td>Update policy by following VLα Compute objective function gradient △i for agent i according to Equation 6</td><td></td></tr><tr><td>Sum gradients Σ △i to update Lα</td><td></td></tr></table>
+
+$V$ , and an auxiliary input $x ( \phi )$ (previously $\pi _ { \phi } .$ ) that can be differentiated with respect to the policy parameters. The latter is critical to be able to differentiate with respect to $\phi$ and in the simplest case it consists of the action as predicted by the policy. While Equation 6 is used for meta-learning $L _ { \alpha }$ , the objective function $L _ { \alpha }$ itself is used for policy learning by following $\nabla _ { \phi } L _ { \alpha } ( \tau , x ( \phi ) , V )$ . See Figure 1 for an overview. MetaGenRL consists of two phases: During meta-training, we alternate between critic updates, objective function updates, and policy updates to meta-learn an objective function $L _ { \alpha }$ as described in Algorithm 1. During meta-testing in Algorithm 2, we take the learned objective function $L _ { \alpha }$ and keep it fixed while training a randomly initialized policy in a new environment to assess its performance.
+
+We note that the inputs to $L _ { \alpha }$ are sampled from a replay buffer rather than solely using on-policy data. If $L _ { \alpha }$ were to represent a REINFORCE-type objective then it would mean that differentiating $L _ { \alpha }$ yields biased policy gradient estimates. In our experiments we will find that the gradients from $L _ { \alpha }$ work much better in comparison to a biased off-policy REINFORCE algorithm, and to an importance-sampled unbiased REINFORCE algorithm, while also improving over the popular on-policy REINFORCE and PPO algorithms.
+
+# 3.2 PARAMETRIZING THE OBJECTIVE FUNCTION
+
+We will implement $L _ { \alpha }$ using an LSTM (Gers et al., 2000; Hochreiter & Schmidhuber, 1997) that iterates over $\tau$ in reverse order and depends on the current policy action $\pi _ { \phi } ( s _ { t } )$ (see Figure 2). At every time-step $L _ { \alpha }$ receives the reward $r _ { t }$ , taken action $a _ { t }$ , predicted action by the current policy $\pi _ { \phi } ( s _ { t } )$ , the time $t$ , and value function estimates $V _ { t } , V _ { t + 1 } { } ^ { 3 }$ . At each step the LSTM outputs the objective value $l _ { t }$ , all of which are summed to yield a single scalar output value that can be differentiated with respect to $\phi$ . In order to accommodate varying action dimensionalities across different environments, both $\pi _ { \phi } ( s _ { t } )$ and $a _ { t }$ are first convolved and then averaged to obtain an action embedding that does not depend on the action dimensionality. Additional details, including suggestions for more expressive alternatives are available in Appendix B.
+
+By presenting the trajectory in reverse order to the LSTM (and $L _ { \alpha }$ correspondingly), it is able to assign credit to an action $a _ { t }$ based on its future impact on the reward, similar to policy gradient estimators. More so, as a general function approximator using these inputs, the LSTM is in principle able to learn different variance and bias reduction techniques, akin to advantage estimates, generalized advantage estimates, or importance weights4. Due to these properties, we expect the class of objective functions that is supported to somewhat relate to a REINFORCE (Williams, 1992) estimator that uses generalized advantage estimation (Schulman et al., 2015b).
+
+<table><tr><td>Algorithm2 MetaGenRL: Meta-Testing</td></tr><tr><td>Require: A test environment e,and an objective function Lα Randomly initialize π,Vθ,B ←</td></tr><tr><td>while f has not converged do if extend replay buffer B then</td></tr><tr><td>Extend Busing π in e</td></tr><tr><td>Sample trajectories from B Update Vθ using TD-error</td></tr></table>
+
+![](images/5ce87b4283df66667aea9cb4e6a804421f831bc4827df80eb82e5e15bc718161.jpg)  
+Figure 2: An overview of $L _ { \alpha } ( \tau , x ( \phi ) , V )$
+
+# 3.3 GENERALITY AND EFFICIENCY OF METAGENRL
+
+MetaGenRL offers a general framework for meta-learning objective functions that can represent a wide range of learning algorithms. In particular, it is only required that both $\pi _ { \phi }$ and $L _ { \alpha }$ can be differentiated w.r.t. to the policy parameters $\phi$ . In the present work, we use this flexibility to leverage population-based meta-optimization, increase sample efficiency through off-policy secondorder gradients, and to improve the generalization capabilities of meta-learned objective functions.
+
+Population-Based A general objective function should be applicable to a wide range of environments and agent parameters. To this extent MetaGenRL is able to leverage the collective experience of multiple agents to perform meta-learning by using a single objective function $L _ { \alpha }$ shared among a population of agents that each act in their own (potentially different) environment. Each agent locally computes Equation 6 over a batch of trajectories, and the resulting gradients are combined to update $L _ { \alpha }$ . Thus, the relevant learning experience of each individual agent is compressed into the objective function that is available to the entire population at any given time.
+
+Sample Efficiency An alternative to learning neural objective functions using a population of agents is through evolution as in EPG (Houthooft et al., 2018). However, we expect meta-learning using second-order gradients as in MetaGenRL to be much more sample efficient. This is due to off-policy training of the objective function $L _ { \alpha }$ and its subsequent off-policy use to improve the policy. Indeed, unlike in evolution there is no need to train multiple randomly initialized agents in their entirety in order to evaluate the objective function, thus speeding up credit assignment. Rather, at any point in time, any information that is deemed useful for future environment interactions can directly be incorporated into the objective function. Finally, using the formulation in Equation 6 one can measure the effects of improving the policy using $L _ { \alpha }$ for multiple steps by increasing the corresponding number of gradient steps before applying $Q _ { \theta }$ , which we will explore in Section 5.2.3.
+
+Meta-Generalization The focus of this work is to learn general learning rules that during testtime can be applied to vastly different environments. A strict separation between the policy and the learning rule, the functional form of the latter, and training across many environments all contribute to this. Regarding the former, a clear separation between the policy and the learning rule as in MetaGenRL is expected to be advantageous for two reasons. Firstly, it allows us to specify the number of parameters of the learning rule independent of the policy and critic parameters. For example, our implementation of $L _ { \alpha }$ uses only $1 5 K$ parameters for the objective function compared to $3 8 4 K$ parameters for the policy and critic. Hence, we are able to only use a short description length for the learning rule. A second advantage that is gained is that the meta-learner is unable to directly change the policy and must, therefore, learn to make use of the objective function. This makes it difficult for the meta-learner to overfit to the training environments.
+
+# 4 RELATED WORK
+
+Among the earliest pursuits in meta-learning are meta-hierarchies of genetic algorithms (Schmidhuber, 1987) and learning update rules in supervised learning (Bengio et al., 1990). While the former introduced a general framework of entire meta-hierarchies, it relied on discrete non-differentiable programs. The latter introduced local update rules that included free parameters, which could be learned using gradients in a supervised setting. Schmidhuber (1993) introduced a differentiable self-referential RNN that could address and modify its own weights, albeit difficult to learn.
+
+Hochreiter et al. (2001) introduced differentiable meta-learning using RNNs to scale to larger problem instances. By giving an RNN access to its prediction error, it could implement its own metalearning algorithm, where the weights are the meta-learned parameters, and the hidden states the subject of learning. This was later extended to the RL setting (Wang et al., 2016; Duan et al., 2016; Santoro et al., 2016; Mishra et al., 2018) (here refered to as $\bar { \mathsf { R L } } ^ { 2 }$ ). As we show empirically in our paper, meta-learning with $\mathtt { R L } ^ { 2 }$ does not generalize well. It lacks a clear separation between policy and objective function, which makes it easy to overfit on training environments. This is exacerbated by the imbalance of $O ( n ^ { 2 } )$ meta-learned parameters to learn $O ( n )$ activations, unlike in MetaGenRL.
+
+Many other recent meta-learning algorithms learn a policy parameter initialization that is later finetuned using a fixed reinforcement learning algorithm (Finn et al., 2017; Schulman et al., 2017; Grant et al., 2018; Yoon et al., 2018). Different from MetaGenRL, these approaches use second order gradients on the same policy parameter vector instead of using a separate objective function. Albeit in principle general (Finn & Levine, 2018), the mixing of policy and learning algorithm leads to a complicated way of expressing general update rules. Similar to $\mathrm { { R L ^ { 2 } } }$ , adaptation to related tasks is possible, but generalization is difficult (Houthooft et al., 2018).
+
+Objective functions have been learned prior to MetaGenRL. Houthooft et al. (2018) evolve an objective function that is later used to train an agent. Unlike MetaGenRL, this approach is extremely costly in terms of the number of environment interactions required to evaluate and update the objective function. Most recently, Bechtle et al. (2019) introduced learned loss functions for reinforcement learning that also make use of second-order gradients, but use a policy gradient estimator instead of a Q-function. Similar to other work, their focus is only on narrow task distributions. Learned objective functions have also been used for learning unsupervised representations (Metz et al., 2019), DDPG-like meta-gradients for hyperparameter search (Xu et al., 2018), and learning from human demonstrations (Yu et al., 2018). Concurrent to our work, Alet et al. (2020) uses techniques from architecture search to search for viable artificial curiosity objectives that are composed of primitive objective functions.
+
+Li & Malik (2016; 2017) and Andrychowicz et al. (2016) conduct meta-learning by learning optimizers that update parameters $\phi$ by modulating the gradient of some fixed objective function $L$ : $\Delta \phi = f _ { \alpha } ( \nabla _ { \phi } L )$ where $\alpha$ is learned. They differ from MetaGenRL in that they only modulate the gradient of a fixed objective function $L$ instead of learning $L$ itself.
+
+Another connection exists to meta-learned intrinsic reward functions (Schmidhuber, 1991a; Dayan & Hinton, 1993; Wiering & Schmidhuber, 1996; Singh et al., 2004; Niekum et al., 2011; Zheng et al., 2018; Jaderberg et al., 2019). Choosing $\begin{array} { r } { \nabla _ { \phi } L _ { \alpha } = \tilde { \nabla _ { \phi } } \sum _ { t = 1 } ^ { T } \bar { r } _ { t } ( \tau ) } \end{array}$ , where $\bar { r } _ { t }$ is a meta-learned reward and $\tilde { \nabla } _ { \theta }$ is a gradient estimator (such as a value based or policy gradient based estimator) reveals that meta-learning objective functions includes meta-learning the gradient estimatior $\tilde { \nabla }$ itself as long as it is expressible by a gradient $\nabla _ { \theta }$ on an objective $L _ { \alpha }$ . In contrast, for intrinsic reward functions, the gradient estimator $\tilde { \nabla }$ is normally fixed.
+
+Finally, we note that positive transfer between different tasks (reward functions) as well as environments (e.g. different Atari games) has been shown previously in the context of transfer learning (Kistler et al., 1997; Parisotto et al., 2015; Rusu et al., 2016; 2019; Nichol et al., 2018) and meta-critic learning across tasks (Sung et al., 2017). In contrast to this work, the approaches that have shown to be successful in this domain rely entirely on human-engineered learning algorithms.
+
+# 5 EXPERIMENTS
+
+We investigate the learning and generalization capabilities of MetaGenRL on several continuous control benchmarks including HalfCheetah (Cheetah) and Hopper from MuJoCo (Todorov et al., 2012), and LunarLanderContinuous (Lunar) from OpenAI gym (Brockman et al., 2016). These environments differ significantly in terms of the properties of the underlying system that is to be controlled, and in terms of the dynamics that have to be learned to complete the environment. Hence, by training meta-RL algorithms on one environment and testing on other environments they provide a reasonable measure of out-of-distribution generalization.
+
+Table 1: Mean return across multiple seeds (MetaGenRL: 6 meta-train $\times ~ 2$ meta-test seeds, $\mathtt { R L } ^ { 2 }$ : 6 meta-train $\times ~ 2$ meta-test seeds, EPG: 3 meta-train $\times ~ 2$ meta-test seeds) obtained by training randomly initialized agents during meta-test time on previously seen environments (cyan) and on unseen environments (brown). Boldface highlights best meta-learned algorithm. Mean returns (6 seeds) of several human-engineered algorithms are also listed.   
+
+<table><tr><td colspan="2">Training \Testing</td><td>Cheetah</td><td>Hopper</td><td>Lunar</td></tr><tr><td>Cheetah &amp; Hopper</td><td>MetaGenRL EPG RL²</td><td>2185 -571 5180</td><td>2439 20 289</td><td>18 -540 -479</td></tr><tr><td>Lunar&amp; Cheetah</td><td>MetaGenRL EPG RL² MetaGenRL (40 agents)</td><td>2552 -701 2218 3106</td><td>2363 8 5 2869</td><td>258 -707 283</td></tr><tr><td>Lunar &amp; Hopper &amp; Walker &amp; Ant Cheetah &amp; Lunar &amp; Walker&amp; Ant Cheetah&amp; Hopper&amp; Walker&amp;Ant</td><td></td><td>3331 2541</td><td>2452 2345</td><td>201 -71 -148</td></tr><tr><td>on-policy REINFORCE (GAE)</td><td>PPO DDPG /TD3 off-policy REINFORCE (GAE)</td><td>1455 8315 -88</td><td>1894 2718 1804</td><td>187 288 168</td></tr></table>
+
+In our experiments, we will mainly compare to EPG and to $\mathrm { { R L ^ { 2 } } }$ to evaluate the efficacy of our approach. We will also compare to several fixed model-free RL algorithms to measure how well the algorithms meta-learned by MetaGenRL compare to these handcrafted alternatives. Unless otherwise mentioned, we will meta-train MetaGenRL using 20 agents that are distributed equally over the indicated training environments5. Meta-learning uses clipped double-Q learning, delayed policy $\&$ objective updates, and target policy smoothing from TD3 (Fujimoto et al., 2018). We will allow for $6 0 0 K$ environment interactions per agent during meta-training and then meta-test the objective function for $1 M$ interactions. Further details are available in Appendix B.
+
+# 5.1 COMPARISON TO PRIOR WORK
+
+Evaluating on previously seen environments We meta-train MetaGenRL on Lunar and compare its ability to train a randomly initialized agent at test-time (i.e. using the learned objective function and keeping it fixed) to DDPG, PPO, and on- and off-policy REINFORCE (both using GAE) across multiple seeds. Figure 3a shows that MetaGenRL markedly outperforms both the REINFORCE baselines and PPO. Compared to DDPG, which finds the optimal policy, MetaGenRL performs only slightly worse on average although the presence of outliers increases its variance. In particular, we find that some meta-test agents get ‘stuck’ for some time before reaching the optimal policy (see Section A.2 for additional analysis). Indeed, when evaluating only the best meta-learned objective function that was obtained during meta-training (MetaGenRL (best objective func) in Figure 3a) we are able to observe a strong reduction in variance and even better performance.
+
+We also report results (Figure 3a) when meta-training MetaGenRL on both Lunar and Cheetah, and compare to EPG and $\Dot { \mathrm { R L ^ { 2 } } }$ that were meta-trained on these same environments6. For MetaGenRL we were able to obtain similar performance to meta-training on only Lunar in this case. In contrast, for EPG it can be observed that even one billion environment interactions is insufficient to find a good objective function (in Figure 3a quickly dropping below -300). Finally, we find that $\mathtt { R L } ^ { 2 }$ reaches the optimal policy after 100 million meta-training iterations, and that its performance is unaffected by additional steps during testing on Lunar. We note that $\mathtt { R L } ^ { 2 }$ does not separate the policy and the learning rule and indeed in a similar ‘within distribution’ evaluation, $\mathtt { R L } ^ { 2 }$ was found successful (Wang et al., 2016; Duan et al., 2016).
+
+![](images/d74d7b1d420935537a14adbd256e7e1b7ea406dd6896c14cd6bc0cec9961df68.jpg)  
+Figure 3: Comparing the test-time training behavior of the meta-learned objective functions by MetaGenRL to other (meta) reinforcement learning algorithms. We train randomly initialized agents on (a) environments that were encountered during training, and (b) on significantly different environments that were unseen. Training environments are denoted by $\dagger$ in the legend. All runs are shown with mean and standard deviation computed over multiple random seeds (MetaGenRL: 6 meta-train $\times 2$ meta-test seeds, $\mathrm { { R L ^ { 2 } } }$ : 6 meta-train $\times 2$ meta-test seeds, EPG: 3 meta-train $\times 2$ meta-test seeds, and 6 seeds for all others).
+
+Table 1 provides a similar comparison for two other environments. Here we find that in general MetaGenRL is able to outperform the REINFORCE baselines and PPO, and in most cases (except for Cheetah) performs similar to $\mathrm { D D P G } ^ { 7 }$ . We also find that MetaGenRL consistently outperforms EPG, and often $\mathtt { R L } ^ { 2 }$ . For an analysis of meta-training on more than two environments we refer to Appendix A.
+
+Generalization to vastly different environments We evaluate the same objective functions learned by MetaGenRL, EPG and the recurrent dynamics by $\mathrm { { R L ^ { 2 } } }$ on Hopper, which is significantly different compared to the meta-training environments. Figure 3b shows that the learned objective function by MetaGenRL continues to outperform both PPO and our implementations of REINFORCE, while the best performing configuration is even able to outperform DDPG.
+
+When comparing to related meta-RL approaches, we find that MetaGenRL is significantly better in this case. The performance of EPG remains poor, which was expected given what was observed on previously seen environments. On the other hand, we now find that the $\mathtt { R L } ^ { 2 }$ baseline fails completely (resulting in a flat low-reward evaluation), suggesting that the learned learning rule that was previously found to be successful is in fact entirely overfitted to the environments that were seen during meta-training. We were able to observe similar results when using different train and test environment splits as reported in Table 1, and in Appendix A.
+
+# 5.2 ANALYSIS
+
+# 5.2.1 META-TRAINING PROGRESSION OF OBJECTIVE FUNCTIONS
+
+Previously we focused on test-time training randomly initialized agents using an objective function that was meta-trained for a total of $6 0 0 K$ steps (corresponding to a total of $1 2 M$ environment interactions across the entire population). We will now investigate the quality of the objective functions during meta-training.
+
+Figure 4 displays the result of evaluating an objective function on Hopper at different intervals during meta-training on Cheetah and Lunar. Initially ( $2 8 K$ steps) it can be seen that due to lack of meta-training there is only a marginal improvement in the return obtained during test time. However, after only meta-training for $8 6 K$ steps we find (perhaps surprisingly) that the meta-trained objective function is already able to make consistent progress in optimizing a randomly initialized agent during test-time. On the other hand, we observe large variances at test-time during this phase of meta-training. Throughout the remaining stages of meta-training we then observe an increase in convergence speed, more stable updates, and a lower variance across seeds.
+
+![](images/323fe16d2eede6436b0b488620e6fa0a8b6b9bce1f803ee1e8a94027c6166c71.jpg)  
+Figure 4: Meta-training with 20 agents on Cheetah and Lunar. We test the objective function at five stages of meta-training by using it to train three randomly initialized agents on Hopper.
+
+![](images/4400a34b33a3a032e0d037e920f21070a525595e7bcbb0e55ae58da6320d551b.jpg)  
+Figure 5: We meta-train MetaGenRL using several alternative parametrizations of $L _ { \alpha }$ on a) Lunar and Cheetah, and b) present results of testing on Cheetah. During meta-training a representative example of a single agent population is shown with shaded regions denoting standard deviation across the population. Meta-test results are reported as per usual across 6 meta-train $\times 2$ meta-test seeds.
+
+# 5.2.2 ABLATION STUDY
+
+We conduct an ablation study of the neural objective function that was described in Section 3.2. In particular, we assess the dependence of $L _ { \alpha }$ on the value estimates $V _ { t } , V _ { t + 1 }$ and on the time component that could to some extent be learned. Other ablations, including limiting access to the action chosen or to the received reward, are expected to be disastrous for generalization to any other environment (or reward function) and therefore not explored.
+
+Dependence on $t$ We use a parameterized objective function of the form $L _ { \alpha } ( a _ { t } , r _ { t } , V _ { t } , \pi _ { \phi } ( s _ { t } ) | t \in$ $0 , . . . , T - 1 )$ as in Figure 2 except that it does not receive information about the time-step $t$ at each step. Although information about the current time-step is required in order to learn (for example) a generalized advantage estimate (Schulman et al., 2015b), the LSTM could in principle learn such time tracking on it own, and we expect only minor effects on meta-training and during meta-testing. Indeed in Figure 5b it can be seen that the neural objective function performs well without access to $t$ , although it converges slower on Cheetah during meta-training (Figure 5a).
+
+Dependence on $V$ We use a parameterized objective function of the form $L _ { \alpha } ( a _ { t } , r _ { t } , t , \pi _ { \phi } ( s _ { t } ) | t \in$ $0 , . . . , T - 1 )$ as in Figure 2 except that it does not receive any information about the value estimates at time-step $t$ . There exist reinforcement learning algorithms that work without value function estimates (eg. Williams (1992); Schmidhuber & Zhao (1998)), although in the absence of an alternative baseline these often have a large variance. Similar results are observed for this ablation in Figure 5a during meta-training where a possibly large variance appears to affect meta-training. Correspondingly during test-time (Figure 5b) we do not find any meaningful training progress to take place. In contrast, we find that we can remove the dependence on one of the value function estimates, i.e. remove $V _ { t + 1 }$ but keep $V _ { t }$ , which during some runs even increases performance.
+
+![](images/a23d51a299e937c5b074bfaeabc1ab47c88c9178b8d5a0c6c170c6614228532d.jpg)  
+Figure 6: We meta-train MetaGenRL on the LunarLander and HalfCheetah environments using one, three, and five inner gradient steps on $\phi$ . Meta-test results are reported across 3 meta-train $\times 2$ meta-test seeds.
+
+# 5.2.3 MULTIPLE GRADIENT STEPS
+
+We analyze the effect of making multiple gradient updates to the policy using $L _ { \alpha }$ before applying the critic to compute second-order gradients with respect to the objective function parameters as in Equation 6. While in previous experiments we have only considered applying a single update, multiple gradient updates might better capture long term effects of the objective function. At the same time, moving further away from the current policy parameters could reduce the overall quality of the second-order gradients. Indeed, in Figure 6 it can be observed that using 3 gradient steps already slightly increases the variance during test-time training on Hopper and Cheetah after metatraining on LunarLander and Cheetah. Similarly, we find that further increasing the number of gradient steps to 5 harms performance.
+
+# 6 CONCLUSION
+
+We have presented MetaGenRL, a novel off-policy gradient-based meta reinforcement learning algorithm that leverages a population of DDPG-like agents to meta-learn general objective functions. Unlike related methods the meta-learned objective functions do not only generalize in narrow task distributions but show similar performance on entirely different tasks while markedly outperforming REINFORCE and PPO. We have argued that this generality is due to MetaGenRL’s explicit separation of the policy and learning rule, the functional form of the latter, and training across multiple agents and environments. Furthermore, the use of second order gradients increases MetaGenRL’s sample efficiency by several orders of magnitude compared to EPG (Houthooft et al., 2018).
+
+In future work, we aim to further improve the learning capabilities of the meta-learned objective functions, including better leveraging knowledge from prior experiences. Indeed, in our current implementation, the objective function is unable to observe the environment or the hidden state of the (recurrent) policy. These extensions are especially interesting as they may allow more complicated curiosity-based (Schmidhuber, 1991b; 1990; Houthooft et al., 2016; Pathak et al., 2017) or model-based (Schmidhuber, 1990; Weber et al., 2017; Ha & Schmidhuber, 2018) algorithms to be learned. To this extent, it will be important to develop introspection methods that analyze the learned objective function and to scale MetaGenRL to make use of many more environments and agents.
+
+# ACKNOWLEDGEMENTS
+
+We thank Paulo Rauber, Imanol Schlag, and the anonymous reviewers for their feedback. This work was supported by the ERC Advanced Grant (no: 742870) and computational resources by the Swiss National Supercomputing Centre (CSCS, project: s978). We also thank NVIDIA Corporation for donating a DGX-1 as part of the Pioneers of AI Research Award and to IBM for donating a Minsky machine.
+
+# REFERENCES
+
+Ferran Alet, Martin F Schneider, Tomas Lozano-Perez, and Leslie Pack Kaelbling. Meta-learning curiosity algorithms. In International Conference on Learning Representations, 2020.
+
+Marcin Andrychowicz, Misha Denil, Sergio Gomez Colmenarejo, Matthew W. Hoffman, David ´ Pfau, Tom Schaul, Brendan Shillingford, and Nando De Freitas. Learning to learn by gradient descent by gradient descent. In Advances in Neural Information Processing Systems, pp. 3988– 3996, 6 2016.
+
+Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer Normalization. arXiv preprint arXiv:1607.06450, 2016.
+
+Sarah Bechtle, Artem Molchanov, Yevgen Chebotar, Edward Grefenstette, Ludovic Righetti, Gaurav Sukhatme, and Franziska Meier. Meta-learning via learned loss. arXiv preprint arXiv:1906.05374, 2019.
+
+Yoshua Bengio, Samy Bengio, and Jocelyn Cloutier. Learning a synaptic learning rule. Universite´ de Montreal, 1990. ´
+
+Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. OpenAI Gym. arXiv preprint arXiv:1606.01540, 2016.
+
+Jeff Clune. AI-GAs: AI-generating algorithms, an alternate paradigm for producing general artificial intelligence. arXiv preprint arXiv:1905.10985, 2019.
+
+P Dayan and G Hinton. Feudal Reinforcement Learning. In D S Lippman, J E Moody, and D S Touretzky (eds.), Advances in Neural Information Processing Systems (NIPS) 5, pp. 271–278. Morgan Kaufmann, 1993.
+
+Yan Duan, John Schulman, Xi Chen, Peter L. Bartlett, Ilya Sutskever, and Pieter Abbeel. RLˆ2: Fast Reinforcement Learning via Slow Reinforcement Learning. arXiv preprint arXiv:1611.02779, 2016.
+
+Chelsea Finn and Sergey Levine. Meta-Learning and Universality: Deep Representations and Gradient Descent can Approximate any Learning Algorithm. In International Conference on Learning Representations, 2018.
+
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In Proceedings of the 34th International Conference on Machine LearningVolume 70, pp. 1126–1135, 2017.
+
+Scott Fujimoto, Herke van Hoof, and David Meger. Addressing function approximation error in actor-critic methods. Proceedings of Machine Learning Research, 80:1587–1596, 2018.
+
+Felix A Gers, Jurgen Schmidhuber, and Fred Cummins. Learning to Forget: Continual Prediction ¨ with LSTM. Neural Computation, 12(10):2451–2471, 2000.
+
+Erin Grant, Chelsea Finn, Sergey Levine, Trevor Darrell, and Thomas Griffiths. Recasting GradientBased Meta-Learning as Hierarchical Bayes. In International Conference on Learning Representations, 2018.
+
+David Ha and Jurgen Schmidhuber. Recurrent world models facilitate policy evolution. In ¨ Advances in Neural Information Processing Systems, pp. 2450–2462, 2018.
+
+S Hochreiter and J Schmidhuber. Long Short-Term Memory. Neural Computation, 9(8):1735–1780, 1997.
+
+Sepp Hochreiter, A. Steven Younger, and Peter R. Conwell. Learning to learn using gradient descent. In International Conference on Artificial Neural Networks, 2001.
+
+Rein Houthooft, Xi Chen, Yan Duan, John Schulman, Filip De Turck, and Pieter Abbeel. VIME: Variational Information Maximizing Exploration. In Advances in Neural Information Processing Systems, pp. 1109–1117, 2016.
+
+Rein Houthooft, Richard Y. Chen, Phillip Isola, Bradly C. Stadie, Filip Wolski, Jonathan Ho, and Pieter Abbeel. Evolved Policy Gradients. In Advances in Neural Information Processing Systems, pp. 5400–5409, 2018.
+
+Max Jaderberg, Wojciech M Czarnecki, Iain Dunning, Luke Marris, Guy Lever, Antonio Garcia Castaneda, Charles Beattie, Neil C Rabinowitz, Ari S Morcos, Avraham Ruderman, Nico- ˜ las Sonnerat, Tim Green, Louise Deason, Joel Z Leibo, David Silver, Demis Hassabis, Koray Kavukcuoglu, and Thore Graepel. Human-level performance in 3D multiplayer games with population-based reinforcement learning. Science (New York, N.Y.), 364(6443):859–865, 5 2019.
+
+Diederik P Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization. In International Conference on Learning Representations, 2015.
+
+Diederik P Kingma and Max Welling. Auto-Encoding Variational Bayes. In International Conference on Learning Representations, 12 2014.
+
+Werner M Kistler, Wulfram Gerstner, and J Leo van Hemmen. Reduction of the Hodgkin-Huxley equations to a single-variable threshold model. Neural Computation, 9(5):1015–1045, 1997.
+
+Ke Li and Jitendra Malik. Learning to Optimize. arXiv preprint arXiv:1606.01885, 2016.
+
+Ke Li and Jitendra Malik. Learning to Optimize Neural Nets. arXiv preprint arXiv:1703.00441, 2017.
+
+Eric Liang, Richard Liaw, Philipp Moritz, Robert Nishihara, Roy Fox, Ken Goldberg, Joseph E Gonzalez, Michael I Jordan, and Ion Stoica. Rllib: Abstractions for distributed reinforcement learning. In International Conference on Machine Learning, pp. 3053–3062, 2018.
+
+Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. In International Conference on Learning Representations, 2016.
+
+Luke Metz, Niru Maheswaranathan, Brian Cheung, and Jascha Sohl-Dickstein. Learning Unsupervised Learning Rules. In International Conference on Learning Representations, 3 2019.
+
+Nikhil Mishra, Mostafa Rohaninejad, and Xi UC Chen Pieter Abbeel Berkeley. A Simple Neural Attentive Meta-Learner. In International Conference on Learning Representations, 2018.
+
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller. Playing Atari with Deep Reinforcement Learning. arXiv preprint arXiv:1312.5602, 2013.
+
+Alex Nichol, Vicki Pfau, Christopher Hesse, Oleg Klimov, and John Schulman Openai. Gotta Learn Fast: A New Benchmark for Generalization in RL. arXiv preprint arXiv:1804.03720, 2018.
+
+Scott Niekum, Lee Spector, and Andrew Barto. Evolution of reward functions for reinforcement learning. In Proceedings of the 13th annual conference companion on Genetic and evolutionary computation, pp. 177–178, 2011.
+
+Emilio Parisotto, Jimmy Lei Ba, and Ruslan Salakhutdinov. Actor-Mimic: Deep Multitask and Transfer Reinforcement Learning. arXiv preprint arXiv:1511.06342, 2015.
+
+Deepak Pathak, Pulkit Agrawal, Alexei A. Efros, and Trevor Darrell. Curiosity-driven exploration by self-supervised prediction. In 34th International Conference on Machine Learning, ICML 2017, volume 6, pp. 4261–4270, 2017. ISBN 9781510855144. doi: 10.1109/CVPRW.2017.70.
+
+Matthias Plappert, Marcin Andrychowicz, Alex Ray, Bob McGrew, Bowen Baker, Glenn Powell, Jonas Schneider, Josh Tobin, Maciek Chociej, Peter Welinder, and others. Multi-goal reinforcement learning: Challenging robotics environments and request for research. arXiv preprint arXiv:1802.09464, 2018.
+
+Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic Backpropagation and Approximate Inference in Deep Generative Models. In International Conference on Machine Learning, pp. 1278–1286, 2014. ISBN 9781634393973. doi: 10.1051/0004-6361/201527329.
+
+Andrei A. Rusu, Sergio Gomez Colmenarejo, Caglar Gulcehre, Guillaume Desjardins, James Kirkpatrick, Razvan Pascanu, Volodymyr Mnih, Koray Kavukcuoglu, and Raia Hadsell. Policy Distillation. In International Conference on Learning Representations, 2016.
+
+Andrei A. Rusu, Dushyant Rao, Jakub Sygnowski, Oriol Vinyals, Razvan Pascanu, Simon Osindero, and Raia Hadsell. Meta-Learning with Latent Embedding Optimization. In International Conference on Learning Representations, 7 2019.
+
+Adam Santoro, Sergey Bartunov, Matthew Botvinick, Daan Wierstra, and Timothy Lillicrap. MetaLearning with Memory-Augmented Neural Networks. In International conference on machine learning, pp. 1842–1850, 2016.
+
+J Schmidhuber. Learning to Generate Sub-Goals for Action Sequences. In T Kohonen, K Makisara, ¨ O Simula, and J Kangas (eds.), Artificial Neural Networks, pp. 967–972. Elsevier Science Publishers B.V., North-Holland, 1991a.
+
+J Schmidhuber. On learning how to learn learning strategies. Technical Report FKI-198-94, Fakultat¨ fur Informatik, Technische Universit ¨ at M¨ unchen, 1994. ¨
+
+J Schmidhuber and J Zhao. Direct policy search and uncertain policy evaluation. Technical Report IDSIA-50-98, IDSIA, Lugano, Switzerland, 1998.
+
+Jurgen Schmidhuber. Evolutionary principles in self-referential learning. Diploma thesis, Institut ¨ fur Informatik, Technische Universit¨ at M¨ unchen, 1987.¨
+
+Jurgen Schmidhuber. Making the world differentiable: On Using Fully Recurrent Self-Supervised ¨ Neural Networks for Dynamic Reinforcement Learning and Planning in Non-Stationary Environments. Technical Report FKI-126-90 (revised), Institut fur Informatik, Technische Universit ¨ at¨ Munchen, 11 1990.¨
+
+Jurgen Schmidhuber. A Possibility for Implementing Curiosity and Boredom in Model-Building ¨ Neural Controllers. In J A Meyer and S W Wilson (eds.), Proc. of the International Conference on Simulation of Adaptive Behavior: From Animals to Animats, pp. 222–227. MIT Press/Bradford Books, 1991b.
+
+Jurgen Schmidhuber. A self-referential weight matrix. In ¨ Proceedings of the International Conference on Artificial Neural Networks, Amsterdam, pp. 446–451. Springer, 1993.
+
+John Schulman, Sergey Levine, Philipp Moritz, Michael I. Jordan, and Pieter Abbeel. Trust Region Policy Optimization. In International conference on machine learning, pp. 1889–1897, 2015a. doi: 10.1063/1.4927398.
+
+John Schulman, Philipp Moritz, Sergey Levine, Michael Jordan, and Pieter Abbeel. HighDimensional Continuous Control Using Generalized Advantage Estimation. arXiv preprint arXiv:1506.02438, 2015b.
+
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal Policy Optimization Algorithms. arXiv preprint arXiv:1707.06347, 2017.
+
+David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In 31st International Conference on Machine Learning, ICML 2014, volume 1, pp. 605–619, 1 2014. ISBN 9781634393973.
+
+Satinder Singh, Satinder Singh, A.G. Barto, A.G. Barto, Nuttapong Chentanez, and Nuttapong Chentanez. Intrinsically motivated reinforcement learning. 18th Annual Conference on Neural Information Processing Systems (NIPS), 17:1281–1288, 2004. ISSN 1943-0604. doi: 10.1109/TAMD.2010.2051031.
+
+Flood Sung, Li Zhang, Tao Xiang, Timothy Hospedales, and Yongxin Yang. Learning to learn: Meta-critic networks for sample efficient learning. arXiv preprint arXiv:1706.09529, 2017.
+
+Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 5026–5033. IEEE, 2012.
+
+Jane X Wang, Zeb Kurth-Nelson, Dhruva Tirumala, Hubert Soyer, Joel Z Leibo, Remi Munos, Charles Blundell, Dharshan Kumaran, and Matt Botvinick. Learning to reinforcement learn. arXiv preprint arXiv:1611.05763, 2016.
+
+Theophane Weber, S ´ ebastien Racani ´ ere, David P. Reichert, Lars Buesing, Arthur Guez, \` Danilo Jimenez Rezende, Adria Puigdomenech Badia, Oriol Vinyals, Nicolas Heess, Yujia Li, \` Razvan Pascanu, Peter Battaglia, David Silver, and Daan Wierstra. Imagination-Augmented Agents for Deep Reinforcement Learning. In Advances in neural information processing systems, pp. 5690–5701, 2017.
+
+M Wiering and J Schmidhuber. HQ-Learning: Discovering Markovian Subgoals for Non-Markovian Reinforcement Learning. Technical Report IDSIA-95-96, IDSIA, 1996.
+
+Daan Wierstra, Tom Schaul, Jan Peters, and Jurgen Schmidhuber. Natural Evolution Strategies. ¨ In 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence), pp. 3381–3387, 2008.
+
+R J Williams. On the Use of Backpropagation in Associative Reinforcement Learning. In IEEE International Conference on Neural Networks, San Diego, volume 2, pp. 263–270, 1988.
+
+Ronald J Williams. Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning. Machine Learning, 8:229–256, 1992.
+
+Zhongwen Xu, Hado Van Hasselt, and David Silver. Meta-gradient reinforcement learning. In Advances in Neural Information Processing Systems, volume 2018-Decem, pp. 2396–2407, 5 2018.
+
+Jaesik Yoon, Taesup Kim, Ousmane Dia, Sungwoong Kim, Yoshua Bengio, and Sungjin Ahn. Bayesian Model-Agnostic Meta-Learning. In Advances in Neural Information Processing Systems, pp. 7332–7342, 2018.
+
+Tianhe Yu, Chelsea Finn, Annie Xie, Sudeep Dasari, Tianhao Zhang, Pieter Abbeel, and Sergey Levine. One-shot imitation from observing humans via domain-adaptive meta-learning. International Conference on Learning Representations, Workshop Track, 2018.
+
+Zeyu Zheng, Junhyuk Oh, and Satinder Singh. On learning intrinsic rewards for policy gradient methods. In Advances in Neural Information Processing Systems, pp. 4644–4654, 2018.
+
+# A ADDITIONAL RESULTS
+
+# A.1 ALL TRAINING AND TEST REGIMES
+
+In the main text, we have shown several combinations of meta-training, and testing environments.   
+We will now show results for all combinations, including the respective human engineered baselines.
+
+![](images/3e41ae1de50cd381df51ba4ea11157c92432a41b05b598e71cf687b8971cbf6c.jpg)  
+Figure 7: Comparing the test-time training behavior of the meta-learned objective functions by MetaGenRL to other (meta) reinforcement learning algorithms on Hopper. We consider within distribution testing (a), and out of distribution testing (b) by varying the meta-training environments (denoted by $\dagger .$ ) for the meta-RL approaches. All runs are shown with mean and standard deviation computed over multiple random seeds (MetaGenRL: 6 meta-train $\times 2$ meta-test seeds, $\mathtt { R L } ^ { 2 }$ : 6 metatrain $\times 2$ meta-test seeds, EPG: 3 meta-train $\times 2$ meta-test seeds, and 6 seeds for all others).
+
+Hopper On Hopper (Figure 7) we find that MetaGenRL works well, both in terms of generalization to previously seen environments, and to unseen environments. The PPO, REINFORCE, $\mathrm { { R L ^ { 2 } } }$ , and EPG baselines are outperformed significantly. Regarding $\mathrm { { R L ^ { 2 } } }$ we observe that it is only able to obtain reward when Hopper was included during meta-training, although its performance is generally poor. Regarding EPG, we observe some learning progress during meta-testing on Hopper after meta-training on Cheetah and Hopper (Figure 7a), although it drops back down quickly as test-time training proceeds. In contrast, when meta-testing on Hopper after meta-training on Cheetah and Lunar (Figure 7b) no test-time training progress is observed at all.
+
+Cheetah Similar results are observed in Figure 8 for Cheetah, where MetaGenRL outperforms PPO and REINFORCE significantly. On the other hand, it can be seen that DDPG notably outperforms MetaGenRL on this environment. It will be interesting to further study these differences in the future to improve the expressibility of our approach. Regarding $\mathrm { { R L ^ { 2 } } }$ and EPG only within distribution generalization results are available due to Cheetah having larger observations and / or action spaces compared to Hopper and Lunar. We observe that $\mathtt { R L } ^ { 2 }$ performs similar to our earlier findings on Hopper but significantly improves in terms of within-distribution generalization (likely due to greater overfitting, as was consistently observed for other splits). EPG shows initially more promise on within distribution generalization (Figure 8a), but ends up like before.
+
+Lunar On Lunar (Figure 9) we find that MetaGenRL is only marginally better compared to the REINFORCE and PPO baselines in terms of within distribution generalization and worse in terms of out of distribution generalization. Analyzing this result reveals that although many of the runs train rather well, some get stuck during the early stages of training without or only delayed recovering. These outliers lead to a seemingly very large variance for MetaGenRL in Figure 9b. We will provide a more detailed analysis of this result in Section A.2. If we focus on the best performing objective function then we observe competitive performance to DDPG (Figure 9a). Nonetheless, we notice that the objective function trained on Hopper generalizes worse to Lunar, despite our earlier result that objective functions trained on Lunar do in fact generalize well to Hopper. MetaGenRL is still able to outperform both $\mathrm { { R L ^ { 2 } } }$ and EPG in terms of out of distribution generalization. We do note that EPG is able to meta-learn objective functions that are able to improve to some extent during test time.
+
+![](images/9a73a7174b731d7e814728462722790fafe0162339853b25ef54b9f45c2950ed.jpg)  
+Figure 8: Comparing the test-time training behavior of the meta-learned objective functions by MetaGenRL to other (meta) reinforcement learning algorithms on Cheetah. We consider within distribution testing (a), and out of distribution testing (b) by varying the meta-training environments (denoted by $\dagger .$ ) for the meta-RL approaches. All runs are shown with mean and standard deviation computed over multiple random seeds (MetaGenRL: 6 meta-train $\times 2$ meta-test seeds, $\mathtt { R L } ^ { 2 }$ : 6 metatrain $\times 2$ meta-test seeds, EPG: 3 meta-train $\times 2$ meta-test seeds, and 6 seeds for all others).
+
+![](images/3839d6ce8b7469decd54610eee04a8f74611de9d754d4f01853f09f30f03c624.jpg)  
+Figure 9: Comparing the test-time training behavior of the meta-learned objective functions by MetaGenRL to other (meta) reinforcement learning algorithms on Lunar. We consider within distribution testing (a), and out of distribution testing (b) by varying the meta-training environments (denoted by $\dagger .$ ) for the meta-RL approaches. All runs are shown with mean and standard deviation computed over multiple random seeds (MetaGenRL: 6 meta-train $\times 2$ meta-test seeds, $\mathrm { { R L ^ { 2 } } }$ : 6 meta-train $\times 2$ meta-test seeds, EPG: 3 meta-train $\times 2$ meta-test seeds, and 6 seeds for all others).
+
+Comparing final scores An overview of the final scores that were obtained for MetaGenRL in comparison to the human engineered baselines is shown in Table 2. It can be seen that MetaGenRL outperforms PPO and off-/on-policy REINFORCE in most configurations while DDPG with TD3 tricks remains stronger on two of the three environments. Note that DDPG is currently not among the representable algorithms by MetaGenRL.
+
+# A.2 STABILITY OF LEARNED OBJECTIVE FUNCTIONS
+
+In the results presented in Figure 9 on Lunar we observed a seemingly large variance for MetaGenRL that was due to outliers. Indeed, when analyzing the individual runs meta-trained on Lunar and tested on Lunar we found that that one of the runs converged to a local optimum early on during training and was unable to recover from this afterwards. On the other hand, we also observed that runs can be ‘stuck’ for a long time to then make very fast learning progress. It suggests that the objective function may sometimes experience difficulties in providing meaningful updates to the policy parameters during the early stages of training.
+
+Table 2: Agent mean return across multiple seeds (MetaGenRL: 6 meta-train $\times 2$ meta-test seeds, and 6 seeds for all others) for meta-test training on previously seen environments (cyan) and on unseen (different) environments (brown) compared to human engineered baselines.   
+
+<table><tr><td></td><td>Training (below)/Test (right)</td><td>Cheetah</td><td>Hopper</td><td>Lunar</td></tr><tr><td>MetaGenRL (20 agents)</td><td>Cheetah &amp; Hopper</td><td>2185</td><td>2433</td><td>18</td></tr><tr><td></td><td>Cheetah &amp; Lunar</td><td>2551</td><td>2363</td><td>258</td></tr><tr><td></td><td>Hopper &amp; Lunar</td><td>4160</td><td>2966</td><td>146</td></tr><tr><td></td><td>Hopper</td><td>3646</td><td>2937</td><td>-62</td></tr><tr><td>MetaGenRL (40 agents)</td><td>Lunar</td><td>4366</td><td> 2717</td><td>244</td></tr><tr><td></td><td>Lunar &amp; Hopper &amp; Walker &amp; Ant</td><td>3106</td><td>2869</td><td>201</td></tr><tr><td></td><td>Cheetah&amp;Lunar&amp;Walker&amp;Ant</td><td>3331</td><td>2452</td><td>-71</td></tr><tr><td></td><td>Cheetah&amp; Hopper&amp; Walker&amp;Ant</td><td>2541</td><td>2345</td><td>-148</td></tr><tr><td>PPO</td><td></td><td>1455</td><td>1894</td><td>187</td></tr><tr><td>DDPG/TD3</td><td></td><td>8315</td><td>2718</td><td>288</td></tr><tr><td>off-policyREINFORCE(GAE)</td><td></td><td>-88</td><td>1804</td><td>168</td></tr><tr><td>on-policy REINFORCE (GAE)</td><td></td><td>38</td><td>565</td><td>120</td></tr></table>
+
+![](images/4b2276764d9244bf736adff14481d2a82ce78d4e685a7e92d18e0603342663f5.jpg)  
+Figure 10: Meta-training with 20 agents on LunarLander. We meta-test the objective function at different stages in training on the same environment.
+
+We have further analyzed this issue by evaluating one of the objective functions at several intervals throughout meta-training in Figure 10. From the meta-training curve (bottom) it can be seen that meta-training in Lunar converges very early. This means that from then on, updates to the objective function will be based on mostly converged policies. As the test-time plots show, these additional updates appear to negatively affect test-time performance. We hypothesize that the objective function essentially ‘forgets’ about the early stages of training a randomly initialized agent, by only incorporating information about good performing agents. A possible solution to this problem would be to keep older policies in the meta-training agent population or use early stopping.
+
+Finally, if we exclude four random seeds (of 12), we indeed find a significant reduction in the variance (and increase in the mean) of the results observed for MetaGenRL (see Figure 11).
+
+# A.3 ABLATION OF AGENT POPULATION SIZE AND UNIQUE ENVIRONMENTS
+
+In our experiments we have used a population of 20 agents during meta-training to ensure diversity in the conditions under which the objective function needs to optimize. The size of this population is a crucial parameter for a stable meta-optimization. Indeed, in Figure 12 it can be seen that metatraining becomes increasingly unstable as the number of agents in the population decreases.
+
+Using a similar argument, one would expect to gain from increasing the number of distinct environments (or agents) during meta-training. In order to verify this, we have evaluated two additional settings: Meta-training on Cheetah & Lunar & Walker & Ant with 20 and 40 agents respectively. Figure 13 shows the result of meta-testing on Hopper for these experiments (also see the final results reported for 40 agents in Table 2). Unexpectedly, we find that increasing the number of distinct environments does not yield a significant improvement and, in fact, sometimes even decrease performance. One possibility is that this is due to the simple form of the objective function under consideration, which has no access to the environment observations to efficiently distinguish between them. Another possibility is that MetaGenRL’s hyperparameters require additional tuning in order to be compatible with these setups.
+
+![](images/05e0df7ed482a21ed24bbbd5c31dd9fbd23ae81e9c51a798b0209c34954cbdb7.jpg)  
+Figure 11: The left plot shows all 12 random seeds on the meta-test environment Lunar while the right has the 4 worst random seeds removed. The variance is now reduced significantly.
+
+![](images/89a2fc338cea3f9c3196149683a9a7c3422be8d61d753a4cb124715b28888c32.jpg)  
+Figure 12: Stable meta-training requires a large population size of at least 20 agents. Metatraining performance is shown for a single run with the mean and standard deviation across the agent population.
+
+![](images/3e6cf636663c178960d7aafafaa73509c21f77171ae6bfa2aaca4bae39f243c5.jpg)  
+Figure 13: Meta-training on Cheetah, Lunar, Walker, and Ant with 20 or 40 agents; metatesting on the out-of-distribution Hopper environment. We compare to previous MetaGenRL configurations.
+
+# B EXPERIMENT DETAILS
+
+In the following we describe all experimental details regarding the architectures used, meta-training, hyperparameters, and baselines. The code to reproduce our experiments is available at http: //louiskirsch.com/code/metagenrl.
+
+# B.1 NEURAL OBJECTIVE FUNCTION ARCHITECTURE
+
+Neural Architecture In this work we use an LSTM to implement the objective function (Figure 2). The LSTM runs backwards in time over the state, action, and reward tuples that were encountered during the trajectory $\tau$ under consideration. At each step $t$ the LSTM receives as input the reward $r _ { t }$ , value estimates of the current and previous state $V _ { t } , V _ { t + 1 }$ , the current timestep $t$ and finally the action that was taken at the current timestep $a _ { t }$ in addition to the action as determined by the current policy $\pi _ { \phi } ( s _ { t } )$ . The actions are first processed by one dimensional convolutional layers striding over the action dimension followed by a reduction to the mean. This allows for different action sizes between environments. Let $A ^ { ( B ) } \in \mathbb { R } ^ { 1 \times D }$ be the action from the replay buffer, $A ^ { ( \pi ) } \in \mathbb { R } ^ { 1 \times D }$ be the action predicted by the policy, and $W \in \mathbb { R } ^ { 2 \times N }$ a learnable matrix corresponding to $N$ outgoing units, then the actions are transformed by
+
+$$
+\frac { 1 } { D } \sum _ { i = 1 } ^ { D } ( [ A ^ { ( B ) } , A ^ { ( \pi ) } ] ^ { T } W ) _ { i } ,
+$$
+
+where $[ a , b ]$ is a concatenation of $a$ and $b$ along the first axis. This corresponds to a convolution with kernel size 1 and stride 1. Further transformations with non-linearities can be added after applying $W$ , if necessary. We found it helpful (but not strictly necessary) to use ReLU activations for half of the units and square activations for the other half.
+
+At each time-step the LSTM outputs a scalar value $l _ { t }$ (bounded between $- \eta$ and $\eta$ using a scaled tanh activation), which are summed to obtain the value of the neural objective function. Differentiating this value with respect to the policy parameters $\phi$ then yields gradients that can be used to improve $\pi _ { \phi }$ . We only allow gradients to flow backwards through $\pi _ { \phi } ( s _ { t } )$ to $\phi$ . This implementation is closely related to the functional form of a REINFORCE (Williams, 1992) estimator using the generalized advantage estimation (Schulman et al., 2015b).
+
+All feed-forward networks (critic and policy) use ReLU activations and layer normalization (Ba et al., 2016). The LSTM uses tanh activations for cell and hidden state transformations, sigmoid activations for the gates. The input time $t$ is normalized between 0 at the beginning of the episode and 1 at the final transition. Any other hyper-parameters can be seen in Table 3.
+
+Extensibility The expressability of the objective function can be further increased through several means. One possibility is to add the entire sequence of state observations $O 1 { : } T$ to its inputs, or by introducing a bi-directional LSTM. Secondly, additional information about the policy (such as the hidden state of a recurrent policy) can be provided to $L$ . Although not explored in this work, this would in principle allow one to learn an objective that encourages certain representations to emerge, e.g. a predictive representation about future observations, akin to a world model (Schmidhuber, 1990; Ha & Schmidhuber, 2018; Weber et al., 2017). In turn, these could create pressure to adapt the policy’s actions to explore unknown dynamics in the environment (Schmidhuber, 1991b; 1990; Houthooft et al., 2016; Pathak et al., 2017).
+
+# B.2 META-TRAINING
+
+Annealing with DDPG At the beginning of meta-training (learning $L _ { \alpha . }$ ), the objective function is randomly initialized and thus does not make sensible updates to the policies. This can lead to irreversibly breaking the policies early during training. Our current implementation circumvents this issue by linearly annealing $\nabla _ { \phi } L _ { \alpha }$ the first 10k timesteps $\sim 2 \%$ of all timesteps) with DDPG $\nabla _ { \phi } Q _ { \theta } \big ( s _ { t } , \pi _ { \phi } \big ( s _ { t } \big ) \big )$ . Preliminary experiments suggested that an exponential learning rate schedule on the gradient of $\nabla _ { \phi } L _ { \alpha }$ for the first 10k steps can replace the annealing with DDPG. The learning rate anneals exponentially between a learning rate of zero and 1e-3. However, in some rare cases this may still lead to unsuccessful training runs, and thus we have omitted this approach from the present work.
+
+Standard training During training, the critic is updated twice as many times as the policy and objective function, similar to TD3 (Fujimoto et al., 2018). One gradient update with data sampled from the replay buffer is applied for every timestep collected from the environment. The gradient with respect to $\phi$ in Equation 6 is combined with $\phi$ using a fixed learning rate in the standard way, all other parameter updates use Adam (Kingma & Ba, 2015) with the default parameters. Any other hyper-parameters can be seen in Table 3 and Table 4.
+
+Using additional gradient steps In our experiments (Section 5.2.3) we analyzed the effect of applying multiple gradient updates to the policy using $L _ { \alpha }$ before applying the critic to compute second-order gradients with respect to the objective function parameters. For two updates, this gives
+
+$$
+\begin{array} { r l } { \nabla _ { \alpha } Q _ { \theta } ( s _ { t } , \pi _ { \phi ^ { \dagger } } ( s _ { t } ) ) \mathrm { ~ w i t h ~ } \phi ^ { \dagger } = \phi ^ { \prime } - \nabla _ { \phi ^ { \prime } } L _ { \alpha } ( \tau _ { 1 } , x ( \phi ^ { \prime } ) , V ) } & { } \\ { \mathrm { ~ a n d ~ } \phi ^ { \prime } = \phi - \nabla _ { \phi } L _ { \alpha } ( \tau _ { 2 } , x ( \phi ) , V ) } & { } \end{array}
+$$
+
+and can be extended to more than two correspondingly. Additionally, we use disjoint mini batches of data $\tau { : } ~ \tau _ { 1 } , \tau _ { 2 }$ . When updating the policy using $\nabla _ { \phi } L _ { \alpha }$ we continue to use only a single gradient step.
+
+# B.3 BASELINES
+
+$\mathbf { R L } ^ { 2 }$ The implementation for $\mathtt { R L } ^ { 2 }$ mimics the paper by Duan et al. (Duan et al., 2016). However, we were unable to achieve good results with TRPO (Schulman et al., 2015a) on the MuJoCo environments and thus used PPO (Schulman et al., 2017) instead. The PPO hyperparameters and implementation are taken from rllib (Liang et al., 2018). Our implementation uses an LSTM with 64 units and does not reset the state of the LSTM for two episodes in sequence. Resetting after additional episodes were given did not improve training results. Different action and observation dimensionalities across environments were handled by using an environment wrapper that pads both with zeros appropriately.
+
+EPG We use the official EPG code base https://github.com/openai/EPG from the original paper (Houthooft et al., 2018). The hyperparameters are taken from the paper, $V = 6 4$ noise vectors, an update frequency of $M = 6 4$ , and 128 updates for every inner loop, resulting in an inner loop length of 8196 steps. During meta-test training, we run with the same update frequency for a total of 1 million steps.
+
+PPO & On-Policy REINFORCE with GAE We use the tuned implementations from https: //spinningup.openai.com/en/latest/spinningup/bench.html which include a GAE (Schulman et al., 2015b) baseline.
+
+Off-Policy Reinforce with GAE The implementation is equivalent to MetaGenRL except that the objective function is fixed to be the REINFORCE estimator with a GAE (Schulman et al., 2015b) baseline. Thus, experience is sampled from a replay buffer. We have also experimented with an importance weighted unbiased estimator but this resulted in poor performance.
+
+DDPG Our implementation is based on https://spinningup.openai.com/en/ latest/spinningup/bench.html and uses the same TD3 tricks (Fujimoto et al., 2018) and hyperparameters (where applicable) that MetaGenRL uses.
+
+Table 3: Architecture hyperparameters   
+
+<table><tr><td>Parameter</td><td> Value</td></tr><tr><td>Critic number of layers</td><td>3</td></tr><tr><td>Critic number of units</td><td>350</td></tr><tr><td>Policy number of layers</td><td>3</td></tr><tr><td>Policy number of units</td><td>350</td></tr><tr><td>Objective function LSTMunits</td><td>32</td></tr><tr><td>Objective function action conv layers</td><td>3</td></tr><tr><td>Objective function action conv filters Error bound n</td><td>32 1000</td></tr></table>
+
+Table 4: Training hyperparameters   
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Truncated episode length Global norm gradient clipping</td><td>20</td></tr><tr><td>Critic learning rate 入1</td><td>1.0 1e-3</td></tr><tr><td>Policy learning rate 入2 Second order learning rate 入3</td><td>1e-3</td></tr><tr><td>Obj. func. learning rate 入4</td><td>1e-3 1e-3</td></tr><tr><td>Critic noise Critic noise clip</td><td>0.2</td></tr><tr><td>Target network update speed Discount factor</td><td>0.5 0.005</td></tr><tr><td>Batch size Random exploration timesteps Policy gaussian noise std Timesteps per agent</td><td>0.99 100 10000 0.1</td></tr></table>

parse/train/SkxgnnNFvH/SkxgnnNFvH.md

@@ -0,0 +1,276 @@
+# Poly-encoders: architectures and pre-training STRATEGIES FOR FAST AND ACCURATE MULTI-SENTENCE SCORING
+
+Samuel Humeau∗, Kurt Shuster∗, Marie-Anne Lachaux, Jason Weston Facebook AI Research {samuelhumeau,kshuster,malachaux,jase}@fb.com
+
+# Abstract
+
+The use of deep pre-trained transformers has led to remarkable progress in a number of applications (Devlin et al., 2019). For tasks that make pairwise comparisons between sequences, matching a given input with a corresponding label, two approaches are common: Cross-encoders performing full self-attention over the pair and $B i$ -encoders encoding the pair separately. The former often performs better, but is too slow for practical use. In this work, we develop a new transformer architecture, the Poly-encoder, that learns global rather than token level self-attention features. We perform a detailed comparison of all three approaches, including what pre-training and fine-tuning strategies work best. We show our models achieve state-of-the-art results on four tasks; that Poly-encoders are faster than Cross-encoders and more accurate than Bi-encoders; and that the best results are obtained by pre-training on large datasets similar to the downstream tasks.
+
+# 1 Introduction
+
+Recently, substantial improvements to state-of-the-art benchmarks on a variety of language understanding tasks have been achieved through the use of deep pre-trained language models followed by fine-tuning (Devlin et al., 2019). In this work we explore improvements to this approach for the class of tasks that require multi-sentence scoring: given an input context, score a set of candidate labels, a setup common in retrieval and dialogue tasks, amongst others. Performance in such tasks has to be measured via two axes: prediction quality and prediction speed, as scoring many candidates can be prohibitively slow.
+
+The current state-of-the-art focuses on using BERT models for pre-training (Devlin et al., 2019), which employ large text corpora on general subjects: Wikipedia and the Toronto Books Corpus (Zhu et al., 2015). Two classes of fine-tuned architecture are typically built on top: Bi-encoders and Cross-encoders. Cross-encoders (Wolf et al., 2019; Vig & Ramea, 2019), which perform full (cross) self-attention over a given input and label candidate, tend to attain much higher accuracies than their counterparts, Bi-encoders (Mazare et al., 2018; Dinan et al., 2019), which perform self-attention ´ over the input and candidate label separately and combine them at the end for a final representation. As the representations are separate, Bi-encoders are able to cache the encoded candidates, and reuse these representations for each input resulting in fast prediction times. Cross-encoders must recompute the encoding for each input and label; as a result, they are prohibitively slow at test time.
+
+In this work, we provide novel contributions that improve both the quality and speed axes over the current state-of-the-art. We introduce the Poly-encoder, an architecture with an additional learnt attention mechanism that represents more global features from which to perform self-attention, resulting in performance gains over Bi-encoders and large speed gains over Cross-Encoders. To pre-train our architectures, we show that choosing abundant data more similar to our downstream task also brings significant gains over BERT pre-training. This is true across all different architecture choices and downstream tasks we try.
+
+We conduct experiments comparing the new approaches, in addition to analysis of what works best for various setups of existing methods, on four existing datasets in the domains of dialogue and information retrieval (IR), with pre-training strategies based on Reddit (Mazare et al., 2018) compared ´ to Wikipedia/Toronto Books (i.e., BERT). We obtain a new state-of-the-art on all four datasets with our best architectures and pre-training strategies, as well as providing practical implementations for real-time use. Our code and models will be released open-source.
+
+# 2 Related Work
+
+The task of scoring candidate labels given an input context is a classical problem in machine learning. While multi-class classification is a special case, the more general task involves candidates as structured objects rather than discrete classes; in this work we consider the inputs and the candidate labels to be sequences of text.
+
+There is a broad class of models that map the input and a candidate label separately into a common feature space wherein typically a dot product, cosine or (parameterized) non-linearity is used to measure their similarity. We refer to these models as $B i$ -encoders. Such methods include vector space models (Salton et al., 1975), LSI (Deerwester et al., 1990), supervised embeddings (Bai et al., 2009; Wu et al., 2018) and classical siamese networks (Bromley et al., 1994). For the next utterance prediction tasks we consider in this work, several Bi-encoder neural approaches have been considered, in particular Memory Networks (Zhang et al., 2018a) and Transformer Memory networks (Dinan et al., 2019) as well as LSTMs (Lowe et al., 2015) and CNNs (Kadlec et al., 2015) which encode input and candidate label separately. A major advantage of Bi-encoder methods is their ability to cache the representations of a large, fixed candidate set. Since the candidate encodings are independent of the input, Bi-encoders are very efficient during evaluation.
+
+Researchers have also studied a more rich class of models we refer to as Cross-encoders, which make no assumptions on the similarity scoring function between input and candidate label. Instead, the concatenation of the input and a candidate serve as a new input to a nonlinear function that scores their match based on any dependencies it wants. This has been explored with Sequential Matching Network CNN-based architectures (Wu et al., 2017), Deep Matching Networks (Yang et al., 2018), Gated Self-Attention (Zhang et al., 2018b), and most recently transformers (Wolf et al., 2019; Vig & Ramea, 2019; Urbanek et al., 2019). For the latter, concatenating the two sequences of text results in applying self-attention at every layer. This yields rich interactions between the input context and the candidate, as every word in the candidate label can attend to every word in the input context, and vice-versa. Urbanek et al. (2019) employed pre-trained BERT models, and fine-tuned both Bi- and Cross-encoders, explicitly comparing them on dialogue and action tasks, and finding that Cross-encoders perform better. However, the performance gains come at a steep computational cost. Cross-encoder representations are much slower to compute, rendering some applications infeasible.
+
+# 3 Tasks
+
+We consider the tasks of sentence selection in dialogue and article search in IR. The former is a task extensively studied and recently featured in two competitions: the Neurips ConvAI2 competition (Dinan et al., 2020), and the DSTC7 challenge, Track 1 (Yoshino et al., 2019; Jonathan K. Kummerfeld & Lasecki, 2018; Chulaka Gunasekara & Lasecki, 2019). We compare on those two tasks and in addition, we also test on the popular Ubuntu V2 corpus (Lowe et al., 2015). For IR, we use the Wikipedia Article Search task of Wu et al. (2018).
+
+The ConvAI2 task is based on the Persona-Chat dataset (Zhang et al., 2018a) which involves dialogues between pairs of speakers. Each speaker is given a persona, which is a few sentences that describe a character they will imitate, e.g. “I love romantic movies”, and is instructed to get to know the other. Models should then condition their chosen response on the dialogue history and the lines of persona. As an automatic metric in the competition, for each response, the model has to pick the correct annotated utterance from a set of 20 choices, where the remaining 19 were other randomly chosen utterances from the evaluation set. Note that in a final system however, one would retrieve from the entire training set of over 100k utterances, but this is avoided for speed reasons in common evaluation setups. The best performing competitor out of 23 entrants in this task achieved $8 0 . 7 \%$ accuracy on the test set utilizing a pre-trained Transformer fine-tuned for this task (Wolf et al., 2019).
+
+The DSTC7 challenge (Track 1) consists of conversations extracted from Ubuntu chat logs, where one partner receives technical support for various Ubuntu-related problems from the other. The best performing competitor (with 20 entrants in Track 1) in this task achieved $6 4 . 5 \%$ R@1 (Chen & Wang, 2019). Ubuntu V2 is a similar but larger popular corpus, created before the competition (Lowe et al., 2015); we report results for this dataset as well, as there are many existing results on it.
+
+Finally, we evaluate on Wikipedia Article Search (Wu et al., 2018). Using the 2016-12-21 dump of English Wikipedia ( $\mathbf { \sigma } \sim 5 \mathbf { M }$ articles), the task is given a sentence from an article as a search query, find the article it came from. Evaluation ranks the true article (minus the sentence) against 10,000 other articles using retrieval metrics. This mimics a web search like scenario where one would like to search for the most relevant articles (web documents). The best reported method is the learningto-rank embedding model, StarSpace, which outperforms fastText, SVMs, and other baselines.
+
+We summarize all four datasets and their statistics in Table 1.
+
+Table 1: Datasets used in this paper.   
+
+<table><tr><td></td><td>ConvAI2</td><td>DTSC7</td><td>Ubuntu V2</td><td>WikiArticleSearch</td></tr><tr><td>Train Ex.</td><td>131,438</td><td>100,000</td><td>1,000.000</td><td>5,035,182</td></tr><tr><td>Valid Ex.</td><td>7,801</td><td>10,000</td><td>19,560</td><td>9,921</td></tr><tr><td>Test Ex.</td><td>6634</td><td>5.000</td><td>18,920</td><td>9,925</td></tr><tr><td>Eval Cands per Ex.</td><td>20</td><td>100</td><td>10</td><td>10,001</td></tr></table>
+
+# 4 Methods
+
+In this section we describe the various models and methods that we explored.
+
+# 4.1 Transformers and Pre-training Strategies
+
+Transformers Our Bi-, Cross-, and Poly-encoders, described in sections 4.2, 4.3 and 4.4 respectively, are based on large pre-trained transformer models with the same architecture and dimension as BERT-base (Devlin et al., 2019), which has 12 layers, 12 attention heads, and a hidden size of 768. As well as considering the BERT pre-trained weights, we also explore our own pre-training schemes. Specifically, we pre-train two more transformers from scratch using the exact same architecture as BERT-base. One uses a similar training setup as in BERT-base, training on 150 million of examples of [INPUT, LABEL] extracted from Wikipedia and the Toronto Books Corpus, while the other is trained on 174 million examples of [INPUT, LABEL] extracted from the online platform Reddit (Mazare et al., 2018), which is a dataset more adapted to dialogue. The former is performed ´ to verify that reproducing a BERT-like setting gives us the same results as reported previously, while the latter tests whether pre-training on data more similar to the downstream tasks of interest helps. For training both new setups we used XLM (Lample & Conneau, 2019).
+
+Input Representation Our pre-training input is the concatenation of input and label [INPUT,LABEL], where both are surrounded with the special token [S], following Lample & Conneau (2019). When pre-training on Reddit, the input is the context, and the label is the next utterance. When pre-training on Wikipedia and Toronto Books, as in Devlin et al. (2019), the input is one sentence and the label the next sentence in the text. Each input token is represented as the sum of three embeddings: the token embedding, the position (in the sequence) embedding and the segment embedding. Segments for input tokens are 0, and for label tokens are 1.
+
+Pre-training Procedure Our pre-training strategy involves training with a masked language model (MLM) task identical to the one in Devlin et al. (2019). In the pre-training on Wikipedia and Toronto Books we add a next-sentence prediction task identical to BERT training. In the pre-training on Reddit, we add a next-utterance prediction task, which is slightly different from the previous one as an utterance can be composed of several sentences. During training $50 \%$ of the time the candidate is the actual next sentence/utterance and $50 \%$ of the time it is a sentence/utterance randomly taken from the dataset. We alternate between batches of the MLM task and the next-sentence/nextutterance prediction task. Like in Lample & Conneau (2019) we use the Adam optimizer with learning rate of 2e-4, $\beta _ { 1 } = 0 . 9$ , $\beta _ { 2 } = 0 . 9 8$ , no L2 weight decay, linear learning rate warmup, and β . β .inverse square root decay of the learning rate. We use a dropout probability of 0.1 on all layers, and a batch of 32000 tokens composed of concatenations [INPUT, LABEL] with similar lengths. We train the model on 32 GPUs for 14 days.
+
+Fine-tuning After pre-training, one can then fine-tune for the multi-sentence selection task of choice, in our case one of the four tasks from Section 3. We consider three architectures with which we fine-tune the transformer: the Bi-encoder, Cross-encoder and newly proposed Poly-encoder.
+
+# 4.2 Bi-encoder
+
+In a Bi-encoder, both the input context and the candidate label are encoded into vectors:
+
+$$
+y _ { c t x t } = r e d ( T _ { 1 } ( c t x t ) ) \qquad y _ { c a n d } = r e d ( T _ { 2 } ( c a n d ) )
+$$
+
+where $T _ { 1 }$ and $T _ { 2 }$ are two transformers that have been pre-trained following the procedure described in 4.1; they initially start with the same weights, but are allowed to update separately during finetuning. $T ( x ) = h _ { 1 } , . . , h _ { N }$ is the output of a transformer $\mathrm { T }$ and $r e d ( \cdot )$ is a function that reduces that , ..,sequence of vectors into one vector. As the input and the label are encoded separately, segment tokens are 0 for both. To resemble what is done during our pre-training, both the input and label are surrounded by the special token [S] and therefore $h _ { 1 }$ corresponds to [S].
+
+We considered three ways of reducing the output into one representation via red(·): choose the first output of the transformer (corresponding to the special token [S]), compute the average over all outputs or the average over the first $m \leq N$ outputs. We compare them in Table 7 in the Appendix. We use the first output of the transformer in our experiments as it gives slightly better results.
+
+Scoring The score of a candidate candi is given by the dot-product $s ( c t x t , c a n d _ { i } ) = y _ { c t x t } \cdot y _ { c a n d _ { i } } ,$ . The network is trained to minimize a cross-entropy loss in which the logits are $y _ { c t x t } \cdot y _ { c a n d _ { 1 } } , . . . , y _ { c t x t } \cdot y _ { c a n d _ { n } }$ , where cand $_ 1$ , ...,is the correct label and the others are chosen from the training set. Similar to Mazare´ et al. (2018), during training we consider the other labels in the batch as negatives. This allows for much faster training, as we can reuse the embeddings computed for each candidate, and also use a larger batch size; e.g., in our experiments on ConvAI2, we were able to use batches of 512 elements.
+
+Inference speed In the setting of retrieval over known candidates, a Bi-encoder allows for the precomputation of the embeddings of all possible candidates of the system. After the context embedding $y _ { c t x t }$ is computed, the only operation remaining is a dot product between $y _ { c t x t }$ and every candidate embedding, which can scale to millions of candidates on a modern GPU, and potentially billions using nearest-neighbor libraries such as FAISS (Johnson et al., 2019).
+
+# 4.3 Cross-encoder
+
+The Cross-encoder allows for rich interactions between the input context and candidate label, as they are jointly encoded to obtain a final representation. Similar to the procedure in pre-training, the context and candidate are surrounded by the special token [S] and concatenated into a single vector, which is encoded using one transformer. We consider the first output of the transformer as the context-candidate embedding:
+
+$$
+y _ { c t x t , c a n d } = h _ { 1 } = f i r s t ( T ( c t x t , c a n d ) )
+$$
+
+where f irst is the function that takes the first vector of the sequence of vectors produced by the transformer. By using a single transformer, the Cross-encoder is able to perform self-attention between the context and candidate, resulting in a richer extraction mechanism than the Bi-encoder. As the candidate label can attend to the input context during the layers of the transformer, the Crossencoder can produce a candidate-sensitive input representation, which the Bi-encoder cannot. For example, this allows it to select useful input features per candidate.
+
+Scoring To score one candidate, a linear layer $W$ is applied to the embedding $y _ { c t x t , c a n d }$ to reduce it from a vector to a scalar:
+
+$$
+s ( c t x t , c a n d _ { i } ) = y _ { c t x t , c a n d _ { i } } W
+$$
+
+Similarly to what is done for the Bi-encoder, the network is trained to minimize a cross entropy loss where the logits are $s ( c t x t , c a n d _ { 1 } ) , . . . , s ( c t x t , c a n d _ { n } )$ , where can $l _ { 1 }$ is the correct candidate and the others are negatives taken from the training set. Unlike in the Bi-encoder, we cannot recycle the other labels of the batch as negatives, so we use external negatives provided in the training set. The Cross-encoder uses much more memory than the Bi-encoder, resulting in a much smaller batch size.
+
+![](images/3110c1d61ef5e5558af249f14c69c29298a158c4e446683e10affa39d7c316ba.jpg)  
+Figure 1: Diagrams of the three model architectures we consider. (a) The Bi-encoder encodes the context and candidate separately, allowing for the caching of candidate representations during inference. (b) The Cross-encoder jointly encodes the context and candidate in a single transformer, yielding richer interactions between context and candidate at the cost of slower computation. (c) The Poly-encoder combines the strengths of the Bi-encoder and Cross-encoder by both allowing for caching of candidate representations and adding a final attention mechanism between global features of the input and a given candidate to give richer interactions before computing a final score.
+
+Inference speed Unfortunately, the Cross-encoder does not allow for precomputation of the candidate embeddings. At inference time, every candidate must be concatenated with the input context and must go through a forward pass of the entire model. Thus, this method cannot scale to a large amount of candidates. We discuss this bottleneck further in Section 5.4.
+
+# 4.4 Poly-encoder
+
+The Poly-encoder architecture aims to get the best of both worlds from the Bi- and Cross-encoder. A given candidate label is represented by one vector as in the Bi-encoder, which allows for caching candidates for fast inference time, while the input context is jointly encoded with the candidate, as in the Cross-encoder, allowing the extraction of more information.
+
+The Poly-encoder uses two separate transformers for the context and label like a Bi-encoder, and the candidate is encoded into a single vector $y _ { c a n d _ { i } }$ . As such, the Poly-encoder method can be implemented using a precomputed cache of encoded responses. However, the input context, which is typically much longer than a candidate, is represented with $m$ vectors $( y _ { c t x t } ^ { 1 } . . . y _ { c t x t } ^ { m } )$ instead of just one as in the Bi-encoder, where $m$ ..will influence the inference speed. To obtain these $m$ global features that represent the input, we learn $m$ context codes $( c _ { 1 } , . . . , c _ { m } )$ , where $c _ { i }$ extracts representation $y _ { c t x t } ^ { i }$ , ...,by attending over all the outputs of the previous layer. That is, we obtain $y _ { c t x t } ^ { i }$ using:
+
+$$
+y _ { c t x t } ^ { i } = \sum _ { j } w _ { j } ^ { c _ { i } } h _ { j } ~ \mathrm { w h e r e } ~ ( w _ { 1 } ^ { c _ { i } } , . . , w _ { N } ^ { c _ { i } } ) = \mathrm { s o f t m a x } ( c _ { i } \cdot h _ { 1 } , . . , c _ { i } \cdot h _ { N } )
+$$
+
+The $m$ context codes are randomly initialized, and learnt during finetuning.
+
+Finally, given our $m$ global context features, we attend over them using $y _ { c a n d _ { i } }$ as the query:
+
+$$
+y _ { c t x t } = \sum _ { i } w _ { i } y _ { c t x t } ^ { i } ~ \mathrm { w h e r e } ~ ( w _ { 1 } , . . , w _ { m } ) = \mathrm { s o f t m a x } ( y _ { c a n d _ { i } } \cdot y _ { c t x t } ^ { 1 } , . . , y _ { c a n d _ { i } } \cdot y _ { c t x t } ^ { m } )
+$$
+
+The final score for that candidate label is then $y _ { c t x t } \cdot y _ { c a n d _ { i } }$ as in a Bi-encoder. As $m < N$ , where $N$ is <the number of tokens, and the context-candidate attention is only performed at the top layer, this is far faster than the Cross-encoder’s full self-attention.
+
+# 5 Experiments
+
+We perform a variety of experiments to test our model architectures and training strategies over four tasks. For metrics, we measure Recall $@ k$ where each test example has $C$ possible candidates to select from, abbreviated to $\operatorname { R @ } k / C$ , as well as mean reciprocal rank (MRR).
+
+# 5.1 Bi-encoders and Cross-encoders
+
+We first investigate fine-tuning the Bi- and Cross-encoder architectures initialized with the weights provided by Devlin et al. (2019), studying the choice of other hyperparameters (we explore our own pre-training schemes in section 5.3). In the case of the Bi-encoder, we can use a large number of negatives by considering the other batch elements as negative training samples, avoiding recomputation of their embeddings. On 8 Nvidia Volta v100 GPUs and using half-precision operations (i.e. float16 operations), we can reach batches of 512 elements on ConvAI2. Table 2 shows that in this setting, we obtain higher performance with a larger batch size, i.e. more negatives, where 511 negatives yields the best results. For the other tasks, we keep the batch size at 256, as the longer sequences in those datasets uses more memory. The Cross-encoder is more computationally intensive, as the embeddings for the (context, candidate) pair must be recomputed each time. We thus limit its batch size to 16 and provide negatives random samples from the training set. For DSTC7 and Ubuntu V2, we choose 15 such negatives; For ConvAI2, the dataset provides 19 negatives.
+
+<table><tr><td rowspan=1 colspan=1>Negatives</td><td rowspan=1 colspan=1>31</td><td rowspan=1 colspan=1>63</td><td rowspan=1 colspan=1>127</td><td rowspan=1 colspan=1>255</td><td rowspan=1 colspan=1>511</td></tr><tr><td rowspan=1 colspan=1>R@1/20</td><td rowspan=1 colspan=1>81.0</td><td rowspan=1 colspan=1>81.7</td><td rowspan=1 colspan=1>82.3</td><td rowspan=1 colspan=1>83.0</td><td rowspan=1 colspan=1>83.3</td></tr></table>
+
+Table 2: Validation performance on ConvAI2 after fine-tuning a Bi-encoder pre-trained with BERT, averaged over 5 runs. The batch size is the number of training negatives $^ { + 1 }$ as we use the other elements of the batch as negatives during training.
+
+The above results are reported with Bi-encoder aggregation based on the first output. Choosing the average over all outputs instead is very similar but slightly worse (83.1, averaged over 5 runs). We also tried to add further non-linearities instead of the inner product of the two representations, but could not obtain improved results over the simpler architecture (results not shown).
+
+We tried two optimizers: Adam (Kingma & Ba, 2015) with weight decay of 0.01 (as recommended by (Devlin et al., 2019)) and Adamax (Kingma & Ba, 2015) without weight decay; based on validation set performance, we choose to fine-tune with Adam when using the BERT weights. The learning rate is initialized to 5e-5 with a warmup of 100 iterations for Bi- and Poly-encoders, and 1000 iterations for the Cross-encoder. The learning rate decays by a factor of 0.4 upon plateau of the loss evaluated on the valid set every half epoch. In Table 3 we show validation performance when fine-tuning various layers of the weights provided by (Devlin et al., 2019), using Adam with decay optimizer. Fine-tuning the entire network is important, with the exception of the word embeddings.
+
+With the setups described above, we fine-tune the Bi- and Cross-encoders on the datasets, and report the results in Table 4. On the first three tasks, our Bi-encoders and Cross-encoders outperform the best existing approaches in the literature when we fine-tune from BERT weights. E.g., the Biencoder reaches $8 1 . 7 \%$ $\mathbb { R } \ @ 1$ on ConvAI2 and $6 6 . 8 \%$ $\mathbb { R } \ @ 1$ on DSTC7, while the Cross-encoder achieves higher scores of $8 4 . 8 \%$ $\mathbb { R } \ @ 1$ on ConvAI2 and $6 7 . 4 \%$ $\mathbf { R } \ @ 1$ on DSTC7. Overall, Crossencoders outperform all previous approaches on the three dialogue tasks, including our Bi-encoders (as expected). We do not report fine-tuning of BERT for Wikipedia IR as we cannot guarantee the test set is not part of the pre-training for that dataset. In addition, Cross-encoders are also too slow to evaluate on the evaluation setup of that task, which has 10k candidates.
+
+Table 3: Validation performance $( \mathbf { R } @ 1 / 2 0 )$ on ConvAI2 using pre-trained weights of BERT-base with different parameters fine-tuned. Average over 5 runs (Bi-encoders) or 3 runs (Cross-encoders).   
+
+<table><tr><td>Fine-tuned parameters</td><td>Bi-encoder</td><td>Cross-encoder</td></tr><tr><td>Top layer</td><td>74.2</td><td>80.6</td></tr><tr><td>Top 4 layers</td><td>82.0</td><td>86.3</td></tr><tr><td>All but Embeddings</td><td>83.3</td><td>87.3</td></tr><tr><td>Every Layer</td><td>83.0</td><td>86.6</td></tr></table>
+
+Table 4: Test performance of Bi-, Poly- and Cross-encoders on our selected tasks.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>ConvAI2</td><td rowspan=1 colspan=2>DSTC7</td><td rowspan=1 colspan=2>Ubuntu v2</td><td rowspan=1 colspan=1>Wikipedia IR</td></tr><tr><td rowspan=1 colspan=1>split</td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=2>test</td><td rowspan=1 colspan=2>test</td><td rowspan=1 colspan=1>test</td></tr><tr><td rowspan=1 colspan=1>metric</td><td rowspan=1 colspan=1>R@1/20</td><td rowspan=1 colspan=1>R@1/100</td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>R@1/10</td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>R@1/10001</td></tr><tr><td rowspan=1 colspan=1>(Wolf et al., 2019)</td><td rowspan=1 colspan=1>80.7</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>(Gu et al., 2018)</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>60.8</td><td rowspan=1 colspan=1>69.1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Chen &amp;Wang,2019)</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>64.5</td><td rowspan=1 colspan=1>73.5</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Yoon et al.,2018)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>65.2</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Dong&amp; Huang,2018)</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>75.9</td><td rowspan=1 colspan=1>84.8</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Wu et al.,2018)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>56.8</td></tr><tr><td rowspan=1 colspan=1>pre-trainedBERTweigh</td><td rowspan=1 colspan=1>tsfrom (De</td><td rowspan=1 colspan=1>linetal.,20</td><td rowspan=1 colspan=1>19)-Toron</td><td rowspan=1 colspan=1>oBooks+</td><td rowspan=1 colspan=1>Vikipedia</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>81.7 ± 0.2</td><td rowspan=1 colspan=1>66.8 ± 0.7</td><td rowspan=1 colspan=1>74.6 ± 0.5</td><td rowspan=1 colspan=1>80.6 ± 0.4</td><td rowspan=1 colspan=1>88.0±0.3</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder16</td><td rowspan=1 colspan=1>83.2 ± 0.1</td><td rowspan=1 colspan=1>67.8 ± 0.3</td><td rowspan=1 colspan=1>75.1 ± 0.2</td><td rowspan=1 colspan=1>81.2 ± 0.2</td><td rowspan=1 colspan=1>88.3± 0.1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 64</td><td rowspan=1 colspan=1>83.7 ± 0.2</td><td rowspan=1 colspan=1>67.0 ± 0.9</td><td rowspan=1 colspan=1>74.7 ± 0.6</td><td rowspan=1 colspan=1>81.3 ± 0.2</td><td rowspan=1 colspan=1>88.4 ± 0.1</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 360</td><td rowspan=1 colspan=1>83.7 ± 0.2</td><td rowspan=1 colspan=1>68.9± 0.4</td><td rowspan=1 colspan=1>76.2 ± 0.2</td><td rowspan=1 colspan=1>80.9± 0.0</td><td rowspan=1 colspan=1>88.1 ± 0.1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>84.8 ± 0.3</td><td rowspan=1 colspan=1>67.4 ± 0.7</td><td rowspan=1 colspan=1>75.6 ± 0.4</td><td rowspan=1 colspan=1>82.8 ± 0.3</td><td rowspan=1 colspan=1>89.4 ± 0.2</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Our pre-training on Toronto Books + </td><td rowspan=1 colspan=1>ntoBooks-</td><td rowspan=1 colspan=1>Wikipedia</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>82.0 ± 0.1</td><td rowspan=1 colspan=1>64.5 ± 0.5</td><td rowspan=1 colspan=1>72.6 ± 0.4</td><td rowspan=1 colspan=1>80.8± 0.5</td><td rowspan=1 colspan=1>88.2 ± 0.4</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 16</td><td rowspan=1 colspan=1>82.7 ± 0.1</td><td rowspan=1 colspan=1>65.3 ± 0.9</td><td rowspan=1 colspan=1>73.2 ± 0.7</td><td rowspan=1 colspan=1>83.4± 0.2</td><td rowspan=1 colspan=1>89.9 ± 0.1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 64</td><td rowspan=1 colspan=1>83.3 ± 0.1</td><td rowspan=1 colspan=1>65.8 ± 0.7</td><td rowspan=1 colspan=1>73.5 ± 0.5</td><td rowspan=1 colspan=1>83.4± 0.1</td><td rowspan=1 colspan=1>89.9± 0.0</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 360</td><td rowspan=1 colspan=1>83.8 ± 0.1</td><td rowspan=1 colspan=1>65.8 ± 0.7</td><td rowspan=1 colspan=1>73.6 ± 0.6</td><td rowspan=1 colspan=1>83.7±0.0</td><td rowspan=1 colspan=1>90.1 ± 0.0</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>84.9 ± 0.3</td><td rowspan=1 colspan=1>65.3 ± 1.0</td><td rowspan=1 colspan=1>73.8± 0.6</td><td rowspan=1 colspan=1>83.1 ± 0.7</td><td rowspan=1 colspan=1>89.7 ± 0.5</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Our pre-training on Reddit</td><td rowspan=1 colspan=1>dit</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>84.8± 0.1</td><td rowspan=1 colspan=1>70.9 ± 0.5</td><td rowspan=1 colspan=1>78.1 ± 0.3</td><td rowspan=1 colspan=1>83.6± 0.7</td><td rowspan=1 colspan=1>90.1 ± 0.4</td><td rowspan=1 colspan=1>71.0</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 16</td><td rowspan=1 colspan=1>86.3 ± 0.3</td><td rowspan=1 colspan=1>71.6 ± 0.6</td><td rowspan=1 colspan=1>78.4± 0.4</td><td rowspan=1 colspan=1>86.0± 0.1</td><td rowspan=1 colspan=1>91.5 ± 0.1</td><td rowspan=1 colspan=1>71.5</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 64</td><td rowspan=1 colspan=1>86.5± 0.2</td><td rowspan=1 colspan=1>71.2 ± 0.8</td><td rowspan=1 colspan=1>78.2 ± 0.7</td><td rowspan=1 colspan=1>85.9± 0.1</td><td rowspan=1 colspan=1>91.5 ± 0.1</td><td rowspan=1 colspan=1>71.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 360</td><td rowspan=1 colspan=1>86.8 ± 0.1</td><td rowspan=1 colspan=1>71.4 ± 1.0</td><td rowspan=1 colspan=1>78.3 ± 0.7</td><td rowspan=1 colspan=1>85.9 ± 0.1</td><td rowspan=1 colspan=1>91.5 ± 0.0</td><td rowspan=1 colspan=1>71.8</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>87.9 ± 0.2</td><td rowspan=1 colspan=1>71.7 ± 0.3</td><td rowspan=1 colspan=1>79.0 ± 0.2</td><td rowspan=1 colspan=1>86.5 ± 0.1</td><td rowspan=1 colspan=1>91.9 ± 0.0</td><td rowspan=1 colspan=1>-</td></tr></table>
+
+# 5.2 Poly-encoders
+
+We train the Poly-encoder using the same batch sizes and optimizer choices as in the Bi-encoder experiments. Results are reported in Table 4 for various values of $m$ context vectors.
+
+The Poly-encoder outperforms the Bi-encoder on all the tasks, with more codes generally yielding larger improvements. Our recommendation is thus to use as large a code size as compute time allows (see Sec. 5.4). On DSTC7, the Poly-encoder architecture with BERT pretraining reaches $6 8 . 9 \%$ R1 with 360 intermediate context codes; this actually outperforms the Cross-encoder result $( 6 7 . 4 \% )$ and is noticeably better than our Bi-encoder result $( 6 6 . 8 \% )$ . Similar conclusions are found on Ubuntu V2 and ConvAI2, although in the latter Cross-encoders give slightly better results.
+
+We note that since reporting our results, the authors of Li et al. (2019) have conducted a human evaluation study on ConvAI2, in which our Poly-encoder architecture outperformed all other models compared against, both generative and retrieval based, including the winners of the competition.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=4>Scoring time (ms)</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>CPU</td><td rowspan=1 colspan=2>GPU</td></tr><tr><td rowspan=1 colspan=1>Candidates</td><td rowspan=1 colspan=1>1k</td><td rowspan=1 colspan=1>100k</td><td rowspan=1 colspan=1>1k</td><td rowspan=1 colspan=1>100k</td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>115</td><td rowspan=1 colspan=1>160</td><td rowspan=1 colspan=1>19</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 16</td><td rowspan=1 colspan=1>122</td><td rowspan=1 colspan=1>678</td><td rowspan=1 colspan=1>18</td><td rowspan=1 colspan=1>38</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 64</td><td rowspan=1 colspan=1>126</td><td rowspan=1 colspan=1>692</td><td rowspan=1 colspan=1>23</td><td rowspan=1 colspan=1>46</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 360</td><td rowspan=1 colspan=1>160</td><td rowspan=1 colspan=1>837</td><td rowspan=1 colspan=1>57</td><td rowspan=1 colspan=1>88</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>21.7k</td><td rowspan=1 colspan=1>2.2M*</td><td rowspan=1 colspan=1>2.6k</td><td rowspan=1 colspan=1>266k*</td></tr></table>
+
+Table 5: Average time in milliseconds to predict the next dialogue utterance from $C$ possible candidates on ConvAI2. \* are inferred.
+
+# 5.3 Domain-specific Pre-training
+
+We fine-tune our Reddit-pre-trained transformer on all four tasks; we additionally fine-tune a transformer that was pre-trained on the same datasets as BERT, specifically Toronto Books $^ +$ Wikipedia. When using our pre-trained weights, we use the Adamax optimizer and optimize all the layers of the transformer including the embeddings. As we do not use weight decay, the weights of the final layer are much larger than those in the final layer of BERT; to avoid saturation of the attention layer in the Poly-encoder, we re-scaled the last linear layer so that the standard deviation of its output matched that of BERT, which we found necessary to achieve good results. We report results of fine-tuning with our pre-trained weights in Table 4. We show that pre-training on Reddit gives further state-ofthe-art performance over our previous results with BERT, a finding that we see for all three dialogue tasks, and all three architectures.
+
+The results obtained with fine-tuning on our own transformers pre-trained on Toronto Books $^ +$ Wikipedia are very similar to those obtained with the original BERT weights, indicating that the choice of dataset used to pre-train the models impacts the final results, not some other detail in our training. Indeed, as the two settings pre-train with datasets of similar size, we can conclude that choosing a pre-training task (e.g. dialogue data) that is similar to the downstream tasks of interest (e.g. dialogue) is a likely explanation for these performance gains, in line with previous results showing multi-tasking with similar tasks is more useful than with dissimilar ones (Caruana, 1997).
+
+# 5.4 Inference Speed
+
+An important motivation for the Poly-encoder architecture is to achieve better results than the Biencoder while also performing at a reasonable speed. Though the Cross-encoder generally yields strong results, it is prohibitively slow. We perform speed experiments to determine the trade-off of improved performance from the Poly-encoder. Specifically, we predict the next utterance for 100 dialogue examples in the ConvAI2 validation set, where the model scores $C$ candidates (in this case, chosen from the training set). We perform these experiments on both CPU-only and GPU setups. CPU computations were run on an 80 core Intel Xeon processor CPU E5-2698. GPU computations were run on a single Nvidia Quadro GP100 using cuda 10.0 and cudnn 7.4.
+
+We show the average time per example for each architecture in Table 5. The difference in timing between the Bi-encoder and the Poly-encoder architectures is rather minimal when there are only 1000 candidates for the model to consider. The difference is more pronounced when considering 100k candidates, a more realistic setup, as we see a 5-6x slowdown for the Poly-encoder variants. Nevertheless, both models are still tractable. The Cross-encoder, however, is 2 orders of magnitude slower than the Bi-encoder and Poly-encoder, rendering it intractable for real-time inference, e.g. when interacting with a dialogue agent, or retrieving from a large set of documents. Thus, Polyencoders, given their desirable performance and speed trade-off, are the preferred method.
+
+We additionally report training times in the Appendix, Table 6. Poly-encoders also have the benefit of being $3 { - } 4 \mathbf { x }$ faster to train than Cross-encoders (and are similar in training time to Bi-encoders).
+
+# 6 Conclusion
+
+In this paper we present new architectures and pre-training strategies for deep bidirectional transformers in candidate selection tasks. We introduced the Poly-encoder method, which provides a mechanism for attending over the context using the label candidate, while maintaining the ability to precompute each candidate’s representation, which allows for fast real-time inference in a production setup, giving an improved trade off between accuracy and speed. We provided an experimental analysis of those trade-offs for Bi-, Poly- and Cross-encoders, showing that Poly-encoders are more accurate than Bi-encoders, while being far faster than Cross-encoders, which are impractical for real-time use. In terms of training these architectures, we showed that pre-training strategies more closely related to the downstream task bring strong improvements. In particular, pre-training from scratch on Reddit allows us to outperform the results we obtain with BERT, a result that holds for all three model architectures and all three dialogue datasets we tried. However, the methods introduced in this work are not specific to dialogue, and can be used for any task where one is scoring a set of candidates, which we showed for an information retrieval task as well.
+
+# References
+
+Bing Bai, Jason Weston, David Grangier, Ronan Collobert, Kunihiko Sadamasa, Yanjun Qi, Olivier Chapelle, and Kilian Weinberger. Supervised semantic indexing. In Proceedings of the 18th ACM conference on Information and knowledge management, pp. 187–196. ACM, 2009.
+
+Jane Bromley, Isabelle Guyon, Yann LeCun, Eduard Sackinger, and Roopak Shah. Signature verifi- ¨ cation using a” siamese” time delay neural network. In Advances in neural information processing systems, pp. 737–744, 1994.
+
+Rich Caruana. Multitask learning. Machine learning, 28(1):41–75, 1997.
+
+Qian Chen and Wen Wang. Sequential attention-based network for noetic end-to-end response selection. CoRR, abs/1901.02609, 2019. URL http://arxiv.org/abs/1901.02609.
+
+Lazaros Polymenakos Chulaka Gunasekara, Jonathan K. Kummerfeld and Walter S. Lasecki. Dstc7 task 1: Noetic end-to-end response selection. In 7th Edition of the Dialog System Technology Challenges at AAAI 2019, January 2019. URL http://workshop.colips.org/dstc7/ papers/dstc7_task1_final_report.pdf.
+
+Scott Deerwester, Susan T Dumais, George W Furnas, Thomas K Landauer, and Richard Harshman. Indexing by latent semantic analysis. Journal of the American society for information science, 41 (6):391–407, 1990.
+
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pp. 4171–4186, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics. doi: 10.18653/v1/N19-1423.
+
+Emily Dinan, Stephen Roller, Kurt Shuster, Angela Fan, Michael Auli, and Jason Weston. Wizard of Wikipedia: Knowledge-powered conversational agents. In Proceedings of the International Conference on Learning Representations (ICLR), 2019.
+
+Emily Dinan, Varvara Logacheva, Valentin Malykh, Alexander Miller, Kurt Shuster, Jack Urbanek, Douwe Kiela, Arthur Szlam, Iulian Serban, Ryan Lowe, Shrimai Prabhumoye, Alan W. Black, Alexander Rudnicky, Jason Williams, Joelle Pineau, Mikhail Burtsev, and Jason Weston. The second conversational intelligence challenge (convai2). In Sergio Escalera and Ralf Herbrich (eds.), The NeurIPS ’18 Competition, pp. 187–208, Cham, 2020. Springer International Publishing. ISBN 978-3-030-29135-8.
+
+Jianxiong Dong and Jim Huang. Enhance word representation for out-of-vocabulary on ubuntu dialogue corpus. CoRR, abs/1802.02614, 2018. URL http://arxiv.org/abs/1802.02614.
+
+Jia-Chen Gu, Zhen-Hua Ling, Yu-Ping Ruan, and Quan Liu. Building sequential inference models for end-to-end response selection. CoRR, abs/1812.00686, 2018. URL http://arxiv.org/ abs/1812.00686.
+
+J. Johnson, M. Douze, and H. Jgou. Billion-scale similarity search with gpus. IEEE Transactions on Big Data, pp. 1–1, 2019. ISSN 2372-2096. doi: 10.1109/TBDATA.2019.2921572.
+
+Joseph Peper Vignesh Athreya Chulaka Gunasekara Jatin Ganhotra Siva Sankalp Patel Lazaros Polymenakos Jonathan K. Kummerfeld, Sai R. Gouravajhala and Walter S. Lasecki. Analyzing assumptions in conversation disentanglement research through the lens of a new dataset and model. ArXiv e-prints, October 2018. URL https://arxiv.org/pdf/1810.11118.pdf.
+
+Rudolf Kadlec, Martin Schmid, and Jan Kleindienst. Improved deep learning baselines for ubuntu corpus dialogs. CoRR, abs/1510.03753, 2015. URL http://arxiv.org/abs/1510.03753.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015. URL http://arxiv.org/abs/1412.6980.
+
+Guillaume Lample and Alexis Conneau. Cross-lingual language model pretraining. Advances in Neural Information Processing Systems (NeurIPS), 2019.
+
+Margaret Li, Jason Weston, and Stephen Roller. Acute-eval: Improved dialogue evaluation with optimized questions and multi-turn comparisons. arXiv preprint arXiv:1909.03087, 2019.
+
+Ryan Lowe, Nissan Pow, Iulian Serban, and Joelle Pineau. The ubuntu dialogue corpus: A large dataset for research in unstructured multi-turn dialogue systems. In SIGDIAL Conference, 2015.
+
+Pierre-Emmanuel Mazare, Samuel Humeau, Martin Raison, and Antoine Bordes. Training millions ´ of personalized dialogue agents. In EMNLP, 2018.
+
+Gerard Salton, Anita Wong, and Chung-Shu Yang. A vector space model for automatic indexing. Communications of the ACM, 18(11):613–620, 1975.
+
+Jack Urbanek, Angela Fan, Siddharth Karamcheti, Saachi Jain, Samuel Humeau, Emily Dinan, Tim Rocktaschel, Douwe Kiela, Arthur Szlam, and Jason Weston. Learning to speak and act in a ¨ fantasy text adventure game. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pp. 673–683, Hong Kong, China, November 2019. Association for Computational Linguistics. doi: 10.18653/v1/D19-1062.
+
+Jesse Vig and Kalai Ramea. Comparison of transfer-learning approaches for response selection in multi-turn conversations. Workshop on DSTC7, 2019.
+
+Thomas Wolf, Victor Sanh, Julien Chaumond, and Clement Delangue. Transfertransfo: A transfer learning approach for neural network based conversational agents. arXiv preprint arXiv:1901.08149, 2019.
+
+Ledell Yu Wu, Adam Fisch, Sumit Chopra, Keith Adams, Antoine Bordes, and Jason Weston. Starspace: Embed all the things! In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.
+
+Yu Ping Wu, Wei Chung Wu, Chen Xing, Ming Zhou, and Zhoujun Li. Sequential matching network: A new architecture for multi-turn response selection in retrieval-based chatbots. In ACL, 2017.
+
+Liu Yang, Minghui Qiu, Chen Qu, Jiafeng Guo, Yongfeng Zhang, W Bruce Croft, Jun Huang, and Haiqing Chen. Response ranking with deep matching networks and external knowledge in information-seeking conversation systems. In The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval, pp. 245–254. ACM, 2018.
+
+Seunghyun Yoon, Joongbo Shin, and Kyomin Jung. Learning to rank question-answer pairs using hierarchical recurrent encoder with latent topic clustering. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), pp. 1575–1584, New Orleans, Louisiana, June 2018. Association for Computational Linguistics. doi: 10.18653/v1/N18-1142.
+
+Koichiro Yoshino, Chiori Hori, Julien Perez, Luis Fernando D’Haro, Lazaros Polymenakos, R. Chulaka Gunasekara, Walter S. Lasecki, Jonathan K. Kummerfeld, Michel Galley, Chris Brockett, Jianfeng Gao, William B. Dolan, Xiang Gao, Huda AlAmri, Tim K. Marks, Devi Parikh, and Dhruv Batra. Dialog system technology challenge 7. CoRR, abs/1901.03461, 2019.
+
+Saizheng Zhang, Emily Dinan, Jack Urbanek, Arthur Szlam, Douwe Kiela, and Jason Weston. Personalizing dialogue agents: I have a dog, do you have pets too? In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics, pp. 2204–2213, Melbourne, Australia, July 2018a. Association for Computational Linguistics.
+
+Zhuosheng Zhang, Jiangtong Li, Pengfei Zhu, Hai Zhao, and Gongshen Liu. Modeling multi-turn conversation with deep utterance aggregation. In COLING, 2018b.
+
+Yukun Zhu, Ryan Kiros, Richard S. Zemel, Ruslan R. Salakhutdinov, Raquel Urtasun, Antonio Torralba, and Sanja Fidler. Aligning books and movies: Towards story-like visual explanations by watching movies and reading books. 2015 IEEE International Conference on Computer Vision (ICCV), pp. 19–27, 2015.
+
+# A Training Time
+
+We report the training time on 8 GPU Volta 100 for the 3 datasets considered and for 4 types of models in Table 6.
+
+Table 6: Training time in hours.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>ConvAI2</td><td rowspan=1 colspan=1>DSTC7</td><td rowspan=1 colspan=1>UbuntuV2</td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>2.0</td><td rowspan=1 colspan=1>4.9</td><td rowspan=1 colspan=1>7.9</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 16</td><td rowspan=1 colspan=1>2.7</td><td rowspan=1 colspan=1>5.5</td><td rowspan=1 colspan=1>8.0</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 64</td><td rowspan=1 colspan=1>2.8</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>8.0</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder64</td><td rowspan=1 colspan=1>9.4</td><td rowspan=1 colspan=1>13.5</td><td rowspan=1 colspan=1>39.9</td></tr></table>
+
+# B Reduction layer in Bi-encoder
+
+We provide in Table 7 the results obtained for different types of reductions on top of the Bi-encoder. Specifically we compare the Recall $@$ 1/20 on the ConvAI2 validation set when taking the first output of BERT, the average of the first 16 outputs, the average of the first 64 outputs and all of them except the first one ([S]).
+
+<table><tr><td rowspan=1 colspan=1>Setup</td><td rowspan=1 colspan=1>ConvAI2 valid Recall@1/20</td></tr><tr><td rowspan=1 colspan=1>First output</td><td rowspan=1 colspan=1>83.3</td></tr><tr><td rowspan=1 colspan=1>Avg first 16 outputs</td><td rowspan=1 colspan=1>82.9</td></tr><tr><td rowspan=1 colspan=1>Avg first 64 outputs</td><td rowspan=1 colspan=1>82.7</td></tr><tr><td rowspan=1 colspan=1>Avg all outputs</td><td rowspan=1 colspan=1>83.1</td></tr></table>
+
+Table 7: Bi-encoder results on the ConvAI2 valid set for different choices of function red(·).
+
+# C Alternative Choices for Context Vectors
+
+We considered a few other ways to derive the context vectors $( y _ { c t x t } ^ { 1 } , . . . , y _ { c t x t } ^ { m } )$ of the Poly-encoder from the output $( h _ { c t x t } ^ { 1 } , . . . , h _ { c t x t } ^ { N } )$ of the underlying transformer:
+
+• Learn $m$ codes $( c _ { 1 } , . . . , c _ { m } )$ , where $c _ { i }$ extracts representation $y _ { c t x t } ^ { i }$ by attending over all the outputs $( h _ { c t x t } ^ { 1 } , . . . , h _ { c t x t } ^ { N } )$ , . This method is denoted “Poly-encoder (Learnt-codes)” or “Poly, ...,encoder (Learnt-m)”, and is the method described in section 4.4   
+• Consider the first $m$ outputs $( h _ { c t x t } ^ { 1 } , . . . , h _ { c t x t } ^ { m } )$ . This method is denoted “Poly-encoder (First $m$ , ..., outputs)” or “Poly-encoder (First-m)”. Note that when $N \ < \ m$ , only $m$ vectors are considered.   
+• Consider the last $m$ outputs.   
+• Consider the last $m$ outputs concatenated with the first one, $h _ { c t x t } ^ { 1 }$ which plays a particular role in BERT as it corresponds to the special token [S].
+
+The performance of those four methods is evaluated on the validation set of Convai2 and DSTC7 and reported on Table 8. The first two methods are shown in Figure 2. We additionally provide the inference time for a given number of candidates coming from the Convai2 dataset on Table 9.
+
+Table 8: Validation and test performance of Poly-encoder variants, with weights initialized from (Devlin et al., 2019). Scores are shown for ConvAI2 and DSTC 7 Track 1. Bold numbers indicate the highest performing variant within that number of codes.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=3>ConvAI2</td><td rowspan=1 colspan=1>DS</td><td rowspan=1 colspan=1>TC7</td></tr><tr><td rowspan=1 colspan=1>split</td><td rowspan=1 colspan=2>dev</td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1>test</td></tr><tr><td rowspan=1 colspan=1>metric</td><td rowspan=1 colspan=2>R@1/20</td><td rowspan=1 colspan=1>R@1/20</td><td rowspan=1 colspan=1>R@1/100</td><td rowspan=1 colspan=1>R@1/100</td></tr><tr><td rowspan=1 colspan=1>(Wolf et al., 2019)</td><td rowspan=1 colspan=2>82.1</td><td rowspan=1 colspan=1>80.7</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Chen &amp; Wang,2019)</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>57.3</td><td rowspan=1 colspan=1>64.5</td></tr><tr><td rowspan=1 colspan=3>1 Attention Code</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Learnt-codesFirst m outputsLast m outputsLast m outputs and hctxt</td><td rowspan=1 colspan=2>81.9 ± 0.383.2 ± 0.282.9 ± 0.1</td><td rowspan=1 colspan=1>81.0 ± 0.181.5 ± 0.181.0 ± 0.11</td><td rowspan=1 colspan=1>56.2 ± 0.156.4 ± 0.356.1 ± 0.41</td><td rowspan=1 colspan=1>66.9 ± 0.766.8 ± 0.767.2 ± 1.11</td></tr><tr><td rowspan=1 colspan=1>4 Attention Codes</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>Learnt-codesFirst m outputsLast m outputsLast m outputs and hctxt</td><td rowspan=2 colspan=2>83.8 ± 0.283.4 ± 0.282.8 ± 0.282.9 ± 0.1</td><td rowspan=2 colspan=1>82.2 ± 0.581.6 ± 0.181.3 ± 0.481.4 ± 0.2</td><td rowspan=1 colspan=1>56.5 ± 0.556.9 ± 0.556.0 ± 0.5</td><td rowspan=2 colspan=1>66.8 ± 0.767.2 ± 1.365.8 ± 0.566.1 ± 0.8</td></tr><tr><td rowspan=1 colspan=1>55.8 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>16 Attention Codes</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=3 colspan=1>Learnt-codesFirst m outputsLast m outputsLast m outputs and hctxt</td><td rowspan=3 colspan=2>84.4 ± 0.185.2 ± 0.183.9 ± 0.283.8 ± 0.3</td><td rowspan=2 colspan=1>83.2 ± 0.183.9 ± 0.282.0 ± 0.4</td><td rowspan=1 colspan=1>57.7 ± 0.2</td><td rowspan=1 colspan=1>67.8 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>56.1 ± 1.756.1 ± 0.3</td><td rowspan=1 colspan=1>66</td></tr><tr><td rowspan=1 colspan=1>81.7 ± 0.3</td><td rowspan=1 colspan=1>56.1 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>64 Attention Codes</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Learnt-codesFirst m outputsLast m outputsLast m outputs and hetxt</td><td rowspan=1 colspan=2>84.9 ± 0.186.0 ± 0.284.9 ± 0.385.0 ± 0.2</td><td rowspan=1 colspan=1>83.7 ± 0.284.2 ± 0.282.9 ± 0.283.2 ± 0.2</td><td rowspan=1 colspan=1>58.3 ± 0.457.7 ± 0.657.0 ± 0.257.3 ± 0.3</td><td rowspan=1 colspan=1>67.0± 0.967.1 ± 0.166.5 ± 0.567.1 ± 0.5</td></tr><tr><td rowspan=1 colspan=1>360 Attention Codes</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=4 colspan=1>Learnt-codesFirst m outputsLast m outputsLast m outputs and hctxt</td><td rowspan=1 colspan=2>85.3 ± 0.3</td><td rowspan=1 colspan=1>83.7 ± 0.2</td><td rowspan=1 colspan=1>57.7 ± 0.3</td><td rowspan=1 colspan=1>68.9 ± 0.4</td></tr><tr><td rowspan=2 colspan=2>86.3 ± 0.186.3 ± 0.1</td><td rowspan=2 colspan=1>84.6 ± 0.384.7 ± 0.3</td><td rowspan=1 colspan=1>58.1 ± 0.4</td><td rowspan=2 colspan=1>66.8 ± 0.768.1 ± 0.5</td></tr><tr><td rowspan=2 colspan=2>86.3 ± 0.186.2 ± 0.3</td><td rowspan=1 colspan=1>86.3 ± 0.1</td><td rowspan=2 colspan=1>84.7 ± 0.384.5 ± 0.4</td><td rowspan=1 colspan=1>58.0 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>58.3 ± 0.4</td><td rowspan=1 colspan=1>68.0 ± 0.8</td></tr></table>
+
+Table 9: Average time in milliseconds to predict the next dialogue utterance from $N$ possible candidates. \* are inferred.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=4>Scoring time (ms)</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>CPU</td><td rowspan=1 colspan=2>GPU</td></tr><tr><td rowspan=1 colspan=1>Candidates</td><td rowspan=1 colspan=1>1k</td><td rowspan=1 colspan=1>100k</td><td rowspan=1 colspan=1>1k</td><td rowspan=1 colspan=1>100k</td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>115</td><td rowspan=1 colspan=1>160</td><td rowspan=1 colspan=1>19</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First m outputs) 16</td><td rowspan=1 colspan=1>119</td><td rowspan=1 colspan=1>551</td><td rowspan=1 colspan=1>17</td><td rowspan=1 colspan=1>37</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First m outputs) 64</td><td rowspan=1 colspan=1>124</td><td rowspan=1 colspan=1>570</td><td rowspan=1 colspan=1>17</td><td rowspan=1 colspan=1>39</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First m outputs) 360</td><td rowspan=1 colspan=1>120</td><td rowspan=1 colspan=1>619</td><td rowspan=1 colspan=1>17</td><td rowspan=1 colspan=1>45</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-codes) 16</td><td rowspan=1 colspan=1>122</td><td rowspan=1 colspan=1>678</td><td rowspan=1 colspan=1>18</td><td rowspan=1 colspan=1>38</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-codes) 64</td><td rowspan=1 colspan=1>126</td><td rowspan=1 colspan=1>692</td><td rowspan=1 colspan=1>23</td><td rowspan=1 colspan=1>46</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-codes) 360</td><td rowspan=1 colspan=1>160</td><td rowspan=1 colspan=1>837</td><td rowspan=1 colspan=1>57</td><td rowspan=1 colspan=1>88</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>21.7k</td><td rowspan=1 colspan=1>2.2M*</td><td rowspan=1 colspan=1>2.6k</td><td rowspan=1 colspan=1>266k*</td></tr></table>
+
+![](images/9221b3a1d779860a34fcba41054b7511ced7276ff5b78615080559f193f9eaa8.jpg)  
+Figure 2: (a) The Bi-encoder (b) The Cross-encoder (c) The Poly-encoder with first m vectors. (d) The Poly-encoder with $m$ learnt codes.
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=2>ConvAI2</td><td rowspan=1 colspan=4>DSTC7</td><td rowspan=1 colspan=4>Ubuntu v2</td></tr><tr><td rowspan=1 colspan=1>split</td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>test</td></tr><tr><td rowspan=1 colspan=1>metric</td><td rowspan=1 colspan=1>R@1/20</td><td rowspan=1 colspan=1>R@1/20</td><td rowspan=1 colspan=1>R@1/100</td><td rowspan=1 colspan=1>R@1/100</td><td rowspan=1 colspan=1>R@10/100</td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>R@1/10</td><td rowspan=1 colspan=1>R@1/10</td><td rowspan=1 colspan=1>R@5/10</td><td rowspan=1 colspan=1>MRR</td></tr><tr><td rowspan=1 colspan=1>HuggingFace(Wolf et al.,2019)</td><td rowspan=1 colspan=1>82.1</td><td rowspan=1 colspan=1>80.7</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Chen&amp;Wang,2019)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>57.3</td><td rowspan=1 colspan=1>64.5</td><td rowspan=1 colspan=1>90.2</td><td rowspan=1 colspan=1>73.5</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td></tr><tr><td rowspan=1 colspan=1>(Dong&amp;Huang,2018)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>75.9</td><td rowspan=1 colspan=1>97.3</td><td rowspan=1 colspan=1>84.8</td></tr><tr><td rowspan=1 colspan=1>pre-trained weights from(Dev</td><td rowspan=1 colspan=1>inetal.,201</td><td rowspan=1 colspan=1>9)-Toronto</td><td rowspan=1 colspan=1>Books+Wi</td><td rowspan=1 colspan=1>tipedia</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>83.3 ± 0.2</td><td rowspan=1 colspan=1>81.7 ± 0.2</td><td rowspan=1 colspan=1>56.5 ± 0.4</td><td rowspan=1 colspan=1>66.8 ± 0.7</td><td rowspan=1 colspan=1>89.0 ± 1.0</td><td rowspan=1 colspan=1>74.6 ± 0.5</td><td rowspan=1 colspan=1>80.9± 0.6</td><td rowspan=1 colspan=1>80.6 ± 0.4</td><td rowspan=1 colspan=1>98.2 ± 0.1</td><td rowspan=1 colspan=1>88.0 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 16</td><td rowspan=1 colspan=1>85.2 ± 0.1</td><td rowspan=1 colspan=1>83.9 ± 0.2</td><td rowspan=1 colspan=1>56.7 ± 0.2</td><td rowspan=1 colspan=1>67.0± 0.9</td><td rowspan=1 colspan=1>88.8±0.3</td><td rowspan=1 colspan=1>74.6± 0.6</td><td rowspan=1 colspan=1>81.7 ± 0.5</td><td rowspan=1 colspan=1>81.4 ± 0.6</td><td rowspan=1 colspan=1>98.2 ± 0.1</td><td rowspan=1 colspan=1>88.5 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder(Learnt-m) 16</td><td rowspan=1 colspan=1>84.4 ± 0.1</td><td rowspan=1 colspan=1>83.2 ± 0.1</td><td rowspan=1 colspan=1>57.7 ± 0.2</td><td rowspan=1 colspan=1>67.8± 0.3</td><td rowspan=1 colspan=1>88.6± 0.2</td><td rowspan=1 colspan=1>75.1 ± 0.2</td><td rowspan=1 colspan=1>81.5±0.1</td><td rowspan=1 colspan=1>81.2 ±0.2</td><td rowspan=1 colspan=1>98.2 ±0.0</td><td rowspan=1 colspan=1>88.3 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder(First-m) 64</td><td rowspan=1 colspan=1>86.0 ± 0.2</td><td rowspan=1 colspan=1>84.2 ± 0.2</td><td rowspan=1 colspan=1>57.1 ± 0.2</td><td rowspan=1 colspan=1>66.9 ± 0.7</td><td rowspan=1 colspan=1>89.1 ± 0.2</td><td rowspan=1 colspan=1>74.7 ± 0.4</td><td rowspan=1 colspan=1>82.2 ± 0.6</td><td rowspan=1 colspan=1>81.9 ± 0.5</td><td rowspan=1 colspan=1>98.4 ±0.0</td><td rowspan=1 colspan=1>88.8± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 64</td><td rowspan=1 colspan=1>84.9 ± 0.1</td><td rowspan=1 colspan=1>83.7 ± 0.2</td><td rowspan=1 colspan=1>58.3 ± 0.4</td><td rowspan=1 colspan=1>67.0± 0.9</td><td rowspan=1 colspan=1>89.2 ± 0.2</td><td rowspan=1 colspan=1>74.7 ± 0.6</td><td rowspan=1 colspan=1>81.8 ± 0.1</td><td rowspan=1 colspan=1>81.3 ± 0.2</td><td rowspan=1 colspan=1>98.2 ± 0.1</td><td rowspan=1 colspan=1>88.4 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 360</td><td rowspan=1 colspan=1>86.3 ± 0.1</td><td rowspan=1 colspan=1>84.6 ± 0.3</td><td rowspan=1 colspan=1>57.8 ± 0.5</td><td rowspan=1 colspan=1>67.0± 0.5</td><td rowspan=1 colspan=1>89.6± 0.9</td><td rowspan=1 colspan=1>75.0± 0.6</td><td rowspan=1 colspan=1>82.7 ± 0.4</td><td rowspan=1 colspan=1>82.2 ± 0.6</td><td rowspan=1 colspan=1>98.4±0.1</td><td rowspan=1 colspan=1>89.0 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 360</td><td rowspan=1 colspan=1>85.3 ± 0.3</td><td rowspan=1 colspan=1>83.7 ±0.2</td><td rowspan=1 colspan=1>57.7 ± 0.3</td><td rowspan=1 colspan=1>68.9 ± 0.4</td><td rowspan=1 colspan=1>89.9 ± 0.5</td><td rowspan=1 colspan=1>76.2 ± 0.2</td><td rowspan=1 colspan=1>81.5 ± 0.1</td><td rowspan=1 colspan=1>80.9 ± 0.1</td><td rowspan=1 colspan=1>98.1 ± 0.0</td><td rowspan=1 colspan=1>88.1 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>87.1 ± 0.1</td><td rowspan=1 colspan=1>84.8± 0.3</td><td rowspan=1 colspan=1>59.4 ± 0.4</td><td rowspan=1 colspan=1>67.4± 0.7</td><td rowspan=1 colspan=1>90.5 ± 0.3</td><td rowspan=1 colspan=1>75.6 ± 0.4</td><td rowspan=1 colspan=1>83.3 ± 0.4</td><td rowspan=1 colspan=1>82.8±0.3</td><td rowspan=1 colspan=1>98.4 ± 0.1</td><td rowspan=1 colspan=1>89.4 ± 0.2</td></tr><tr><td rowspan=1 colspan=1>Our pre-training on Toronto Books + Wikipedia</td><td rowspan=1 colspan=1>oks+Wiki</td><td rowspan=1 colspan=1>edia</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>84.6 ± 0.1</td><td rowspan=1 colspan=1>82.0 ± 0.1</td><td rowspan=1 colspan=1>54.9 ± 0.5</td><td rowspan=1 colspan=1>64.5 ± 0.5</td><td rowspan=1 colspan=1>88.1 ± 0.2</td><td rowspan=1 colspan=1>72.6 ± 0.4</td><td rowspan=1 colspan=1>80.9 ± 0.5</td><td rowspan=1 colspan=1>80.8 ± 0.5</td><td rowspan=1 colspan=1>98.4 ± 0.1</td><td rowspan=1 colspan=1>88.2 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 16</td><td rowspan=1 colspan=1>84.1 ± 0.2</td><td rowspan=1 colspan=1>81.4 ± 0.2</td><td rowspan=1 colspan=1>53.9 ± 2.7</td><td rowspan=1 colspan=1>63.3 ± 2.9</td><td rowspan=1 colspan=1>87.2 ± 1.5</td><td rowspan=1 colspan=1>71.6 ± 2.4</td><td rowspan=1 colspan=1>80.8 ± 0.5</td><td rowspan=1 colspan=1>80.6 ±0.4</td><td rowspan=1 colspan=1>98.4 ± 0.1</td><td rowspan=1 colspan=1>88.1 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 16</td><td rowspan=1 colspan=1>85.4± 0.2</td><td rowspan=1 colspan=1>82.7 ±0.1</td><td rowspan=1 colspan=1>56.0± 0.4</td><td rowspan=1 colspan=1>65.3± 0.9</td><td rowspan=1 colspan=1>88.2± 0.7</td><td rowspan=1 colspan=1>73.2 ± 0.7</td><td rowspan=1 colspan=1>84.0± 0.1</td><td rowspan=1 colspan=1>83.4 ±0.2</td><td rowspan=1 colspan=1>98.7±0.0</td><td rowspan=1 colspan=1>89.9 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder(First-m) 64</td><td rowspan=1 colspan=1>86.1 ±0.4</td><td rowspan=1 colspan=1>83.9± 0.3</td><td rowspan=1 colspan=1>55.6 ± 0.9</td><td rowspan=1 colspan=1>64.3 ± 1.5</td><td rowspan=1 colspan=1>87.8 ± 0.4</td><td rowspan=1 colspan=1>72.5 ± 1.0</td><td rowspan=1 colspan=1>80.9 ± 0.6</td><td rowspan=1 colspan=1>80.7 ± 0.6</td><td rowspan=1 colspan=1>98.4± 0.0</td><td rowspan=1 colspan=1>88.2 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 64</td><td rowspan=1 colspan=1>85.6 ± 0.1</td><td rowspan=1 colspan=1>83.3 ± 0.1</td><td rowspan=1 colspan=1>56.2 ± 0.4</td><td rowspan=1 colspan=1>65.8 ± 0.7</td><td rowspan=1 colspan=1>88.4 ± 0.3</td><td rowspan=1 colspan=1>73.5 ± 0.5</td><td rowspan=1 colspan=1>84.0 ± 0.1</td><td rowspan=1 colspan=1>83.4 ± 0.1</td><td rowspan=1 colspan=1>98.7 ±0.0</td><td rowspan=1 colspan=1>89.9 ± 0.0</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 360</td><td rowspan=1 colspan=1>86.6± 0.3</td><td rowspan=1 colspan=1>84.4 ± 0.2</td><td rowspan=1 colspan=1>57.5 ± 0.4</td><td rowspan=1 colspan=1>66.5 ± 1.2</td><td rowspan=1 colspan=1>89.0 ± 0.5</td><td rowspan=1 colspan=1>74.4 ± 0.7</td><td rowspan=1 colspan=1>81.3±0.6</td><td rowspan=1 colspan=1>81.1 ±0.4</td><td rowspan=1 colspan=1>98.4 ± 0.2</td><td rowspan=1 colspan=1>88.4 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder(Learnt-m) 360</td><td rowspan=1 colspan=1>86.1 ± 0.1</td><td rowspan=1 colspan=1>83.8 ± 0.1</td><td rowspan=1 colspan=1>56.5 ± 0.8</td><td rowspan=1 colspan=1>65.8 ± 0.7</td><td rowspan=1 colspan=1>88.5 ± 0.6</td><td rowspan=1 colspan=1>73.6 ± 0.6</td><td rowspan=1 colspan=1>84.2 ± 0.2</td><td rowspan=1 colspan=1>83.7 ±0.0</td><td rowspan=1 colspan=1>98.7 ± 0.1</td><td rowspan=1 colspan=1>90.1 ± 0.0</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>87.3 ± 0.5</td><td rowspan=1 colspan=1>84.9 ± 0.3</td><td rowspan=1 colspan=1>57.7 ± 0.5</td><td rowspan=1 colspan=1>65.3 ± 1.0</td><td rowspan=1 colspan=1>89.7 ± 0.5</td><td rowspan=1 colspan=1>73.8 ± 0.6</td><td rowspan=1 colspan=1>83.2 ±0.8</td><td rowspan=1 colspan=1>83.1 ± 0.7</td><td rowspan=1 colspan=1>98.7 ± 0.1</td><td rowspan=1 colspan=1>89.7 ± 0.5</td></tr><tr><td rowspan=1 colspan=1>Our pre-training on Reddit</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>86.9 ± 0.1</td><td rowspan=1 colspan=1>84.8 ± 0.1</td><td rowspan=1 colspan=1>60.1 ± 0.4</td><td rowspan=1 colspan=1>70.9 ± 0.5</td><td rowspan=1 colspan=1>90.6 ± 0.3</td><td rowspan=1 colspan=1>78.1 ± 0.3</td><td rowspan=1 colspan=1>83.7±0.7</td><td rowspan=1 colspan=1>83.6 ±0.7</td><td rowspan=1 colspan=1>98.8±0.1</td><td rowspan=1 colspan=1>90.1 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder(First-m) 16</td><td rowspan=1 colspan=1>89.0± 0.1</td><td rowspan=1 colspan=1>86.4± 0.3</td><td rowspan=1 colspan=1>60.4± 0.3</td><td rowspan=1 colspan=1>70.7 ± 0.7</td><td rowspan=1 colspan=1>91.0± 0.4</td><td rowspan=1 colspan=1>78.0± 0.5</td><td rowspan=1 colspan=1>84.3 ± 0.3</td><td rowspan=1 colspan=1>84.3± 0.2</td><td rowspan=1 colspan=1>98.9± 0.0</td><td rowspan=1 colspan=1>90.5 ±0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 16</td><td rowspan=1 colspan=1>88.6 ± 0.3</td><td rowspan=1 colspan=1>86.3 ± 0.3</td><td rowspan=1 colspan=1>61.1 ± 0.4</td><td rowspan=1 colspan=1>71.6 ± 0.6</td><td rowspan=1 colspan=1>91.3 ± 0.3</td><td rowspan=1 colspan=1>78.4 ± 0.4</td><td rowspan=1 colspan=1>86.1 ± 0.1</td><td rowspan=1 colspan=1>86.0 ± 0.1</td><td rowspan=1 colspan=1>99.0 ± 0.1</td><td rowspan=1 colspan=1>91.5 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 64</td><td rowspan=1 colspan=1>89.5 ± 0.1</td><td rowspan=1 colspan=1>87.3 ± 0.2</td><td rowspan=1 colspan=1>61.0 ± 0.4</td><td rowspan=1 colspan=1>70.9 ± 0.6</td><td rowspan=1 colspan=1>91.5 ± 0.5</td><td rowspan=1 colspan=1>78.0± 0.3</td><td rowspan=1 colspan=1>84.0 ± 0.4</td><td rowspan=1 colspan=1>83.9 ± 0.4</td><td rowspan=1 colspan=1>98.8±0.0</td><td rowspan=1 colspan=1>90.3 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 64</td><td rowspan=1 colspan=1>89.0± 0.1</td><td rowspan=1 colspan=1>86.5 ± 0.2</td><td rowspan=1 colspan=1>60.9± 0.6</td><td rowspan=1 colspan=1>71.2 ± 0.8</td><td rowspan=1 colspan=1>91.3± 0.4</td><td rowspan=1 colspan=1>78.2± 0.7</td><td rowspan=1 colspan=1>86.2 ± 0.1</td><td rowspan=1 colspan=1>85.9 ± 0.1</td><td rowspan=1 colspan=1>99.1 ± 0.0</td><td rowspan=1 colspan=1>91.5 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 360</td><td rowspan=1 colspan=1>90.0 ± 0.1</td><td rowspan=1 colspan=1>87.3 ± 0.1</td><td rowspan=1 colspan=1>61.1 ± 1.9</td><td rowspan=1 colspan=1>70.9 ± 2.1</td><td rowspan=1 colspan=1>91.5 ± 0.9</td><td rowspan=1 colspan=1>77.9 ± 1.6</td><td rowspan=1 colspan=1>84.8 ± 0.5</td><td rowspan=1 colspan=1>84.6 ± 0.5</td><td rowspan=1 colspan=1>98.9 ± 0.1</td><td rowspan=1 colspan=1>90.7 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 360</td><td rowspan=1 colspan=1>89.2 ± 0.1</td><td rowspan=1 colspan=1>86.8± 0.1</td><td rowspan=1 colspan=1>61.2 ± 0.2</td><td rowspan=1 colspan=1>71.4 ± 1.0</td><td rowspan=1 colspan=1>91.1 ± 0.3</td><td rowspan=1 colspan=1>78.3± 0.7</td><td rowspan=1 colspan=1>86.3 ± 0.1</td><td rowspan=1 colspan=1>85.9 ± 0.1</td><td rowspan=1 colspan=1>99.1 ±0.0</td><td rowspan=1 colspan=1>91.5 ± 0.0</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>90.3± 0.2</td><td rowspan=1 colspan=1>87.9 ± 0.2</td><td rowspan=1 colspan=1>63.9 ± 0.3</td><td rowspan=1 colspan=1>71.7 ± 0.3</td><td rowspan=1 colspan=1>92.4 ± 0.5</td><td rowspan=1 colspan=1>79.0 ± 0.2</td><td rowspan=1 colspan=1>86.7±0.1</td><td rowspan=1 colspan=1>86.5 ± 0.1</td><td rowspan=1 colspan=1>99.1 ± 0.0</td><td rowspan=1 colspan=1>91.9 ± 0.0</td></tr></table>
+
+Table 10: Validation and test performances of Bi-, Poly- and Cross-encoders. Scores are shown for ConvAI2, DSTC7 Track 1 and Ubuntu v2, and the previous state-of-the-art models in the literature.

parse/train/SkxgnnNFvH/SkxgnnNFvH_content_list.json

@@ -0,0 +1,1512 @@
+[
+    {
+        "type": "text",
+        "text": "Poly-encoders: architectures and pre-training STRATEGIES FOR FAST AND ACCURATE MULTI-SENTENCE SCORING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            815,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Samuel Humeau∗, Kurt Shuster∗, Marie-Anne Lachaux, Jason Weston Facebook AI Research {samuelhumeau,kshuster,malachaux,jase}@fb.com ",
+        "bbox": [
+            181,
+            169,
+            674,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            248,
+            535,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The use of deep pre-trained transformers has led to remarkable progress in a number of applications (Devlin et al., 2019). For tasks that make pairwise comparisons between sequences, matching a given input with a corresponding label, two approaches are common: Cross-encoders performing full self-attention over the pair and $B i$ -encoders encoding the pair separately. The former often performs better, but is too slow for practical use. In this work, we develop a new transformer architecture, the Poly-encoder, that learns global rather than token level self-attention features. We perform a detailed comparison of all three approaches, including what pre-training and fine-tuning strategies work best. We show our models achieve state-of-the-art results on four tasks; that Poly-encoders are faster than Cross-encoders and more accurate than Bi-encoders; and that the best results are obtained by pre-training on large datasets similar to the downstream tasks. ",
+        "bbox": [
+            233,
+            280,
+            764,
+            446
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            473,
+            308,
+            489
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, substantial improvements to state-of-the-art benchmarks on a variety of language understanding tasks have been achieved through the use of deep pre-trained language models followed by fine-tuning (Devlin et al., 2019). In this work we explore improvements to this approach for the class of tasks that require multi-sentence scoring: given an input context, score a set of candidate labels, a setup common in retrieval and dialogue tasks, amongst others. Performance in such tasks has to be measured via two axes: prediction quality and prediction speed, as scoring many candidates can be prohibitively slow. ",
+        "bbox": [
+            174,
+            505,
+            825,
+            602
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The current state-of-the-art focuses on using BERT models for pre-training (Devlin et al., 2019), which employ large text corpora on general subjects: Wikipedia and the Toronto Books Corpus (Zhu et al., 2015). Two classes of fine-tuned architecture are typically built on top: Bi-encoders and Cross-encoders. Cross-encoders (Wolf et al., 2019; Vig & Ramea, 2019), which perform full (cross) self-attention over a given input and label candidate, tend to attain much higher accuracies than their counterparts, Bi-encoders (Mazare et al., 2018; Dinan et al., 2019), which perform self-attention ´ over the input and candidate label separately and combine them at the end for a final representation. As the representations are separate, Bi-encoders are able to cache the encoded candidates, and reuse these representations for each input resulting in fast prediction times. Cross-encoders must recompute the encoding for each input and label; as a result, they are prohibitively slow at test time. ",
+        "bbox": [
+            174,
+            609,
+            825,
+            748
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we provide novel contributions that improve both the quality and speed axes over the current state-of-the-art. We introduce the Poly-encoder, an architecture with an additional learnt attention mechanism that represents more global features from which to perform self-attention, resulting in performance gains over Bi-encoders and large speed gains over Cross-Encoders. To pre-train our architectures, we show that choosing abundant data more similar to our downstream task also brings significant gains over BERT pre-training. This is true across all different architecture choices and downstream tasks we try. ",
+        "bbox": [
+            174,
+            755,
+            823,
+            852
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments comparing the new approaches, in addition to analysis of what works best for various setups of existing methods, on four existing datasets in the domains of dialogue and information retrieval (IR), with pre-training strategies based on Reddit (Mazare et al., 2018) compared ´ to Wikipedia/Toronto Books (i.e., BERT). We obtain a new state-of-the-art on all four datasets with our best architectures and pre-training strategies, as well as providing practical implementations for real-time use. Our code and models will be released open-source. ",
+        "bbox": [
+            176,
+            859,
+            823,
+            901
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            167,
+            320,
+            184
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The task of scoring candidate labels given an input context is a classical problem in machine learning. While multi-class classification is a special case, the more general task involves candidates as structured objects rather than discrete classes; in this work we consider the inputs and the candidate labels to be sequences of text. ",
+        "bbox": [
+            174,
+            200,
+            823,
+            256
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "There is a broad class of models that map the input and a candidate label separately into a common feature space wherein typically a dot product, cosine or (parameterized) non-linearity is used to measure their similarity. We refer to these models as $B i$ -encoders. Such methods include vector space models (Salton et al., 1975), LSI (Deerwester et al., 1990), supervised embeddings (Bai et al., 2009; Wu et al., 2018) and classical siamese networks (Bromley et al., 1994). For the next utterance prediction tasks we consider in this work, several Bi-encoder neural approaches have been considered, in particular Memory Networks (Zhang et al., 2018a) and Transformer Memory networks (Dinan et al., 2019) as well as LSTMs (Lowe et al., 2015) and CNNs (Kadlec et al., 2015) which encode input and candidate label separately. A major advantage of Bi-encoder methods is their ability to cache the representations of a large, fixed candidate set. Since the candidate encodings are independent of the input, Bi-encoders are very efficient during evaluation. ",
+        "bbox": [
+            174,
+            263,
+            825,
+            416
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Researchers have also studied a more rich class of models we refer to as Cross-encoders, which make no assumptions on the similarity scoring function between input and candidate label. Instead, the concatenation of the input and a candidate serve as a new input to a nonlinear function that scores their match based on any dependencies it wants. This has been explored with Sequential Matching Network CNN-based architectures (Wu et al., 2017), Deep Matching Networks (Yang et al., 2018), Gated Self-Attention (Zhang et al., 2018b), and most recently transformers (Wolf et al., 2019; Vig & Ramea, 2019; Urbanek et al., 2019). For the latter, concatenating the two sequences of text results in applying self-attention at every layer. This yields rich interactions between the input context and the candidate, as every word in the candidate label can attend to every word in the input context, and vice-versa. Urbanek et al. (2019) employed pre-trained BERT models, and fine-tuned both Bi- and Cross-encoders, explicitly comparing them on dialogue and action tasks, and finding that Cross-encoders perform better. However, the performance gains come at a steep computational cost. Cross-encoder representations are much slower to compute, rendering some applications infeasible. ",
+        "bbox": [
+            174,
+            424,
+            825,
+            603
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 Tasks ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            626,
+            251,
+            642
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We consider the tasks of sentence selection in dialogue and article search in IR. The former is a task extensively studied and recently featured in two competitions: the Neurips ConvAI2 competition (Dinan et al., 2020), and the DSTC7 challenge, Track 1 (Yoshino et al., 2019; Jonathan K. Kummerfeld & Lasecki, 2018; Chulaka Gunasekara & Lasecki, 2019). We compare on those two tasks and in addition, we also test on the popular Ubuntu V2 corpus (Lowe et al., 2015). For IR, we use the Wikipedia Article Search task of Wu et al. (2018). ",
+        "bbox": [
+            174,
+            659,
+            823,
+            742
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The ConvAI2 task is based on the Persona-Chat dataset (Zhang et al., 2018a) which involves dialogues between pairs of speakers. Each speaker is given a persona, which is a few sentences that describe a character they will imitate, e.g. “I love romantic movies”, and is instructed to get to know the other. Models should then condition their chosen response on the dialogue history and the lines of persona. As an automatic metric in the competition, for each response, the model has to pick the correct annotated utterance from a set of 20 choices, where the remaining 19 were other randomly chosen utterances from the evaluation set. Note that in a final system however, one would retrieve from the entire training set of over 100k utterances, but this is avoided for speed reasons in common evaluation setups. The best performing competitor out of 23 entrants in this task achieved $8 0 . 7 \\%$ accuracy on the test set utilizing a pre-trained Transformer fine-tuned for this task (Wolf et al., 2019). ",
+        "bbox": [
+            174,
+            750,
+            823,
+            888
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The DSTC7 challenge (Track 1) consists of conversations extracted from Ubuntu chat logs, where one partner receives technical support for various Ubuntu-related problems from the other. The best performing competitor (with 20 entrants in Track 1) in this task achieved $6 4 . 5 \\%$ R@1 (Chen & Wang, 2019). Ubuntu V2 is a similar but larger popular corpus, created before the competition (Lowe et al., 2015); we report results for this dataset as well, as there are many existing results on it. ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Finally, we evaluate on Wikipedia Article Search (Wu et al., 2018). Using the 2016-12-21 dump of English Wikipedia ( $\\mathbf { \\sigma } \\sim 5 \\mathbf { M }$ articles), the task is given a sentence from an article as a search query, find the article it came from. Evaluation ranks the true article (minus the sentence) against 10,000 other articles using retrieval metrics. This mimics a web search like scenario where one would like to search for the most relevant articles (web documents). The best reported method is the learningto-rank embedding model, StarSpace, which outperforms fastText, SVMs, and other baselines. ",
+        "bbox": [
+            174,
+            151,
+            825,
+            236
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We summarize all four datasets and their statistics in Table 1. ",
+        "bbox": [
+            174,
+            243,
+            575,
+            257
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/1f40734c36a3cb0d6b856eeb0c49308b21f5cad56bd66fab1630c8820705b0cb.jpg",
+        "table_caption": [
+            "Table 1: Datasets used in this paper. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>ConvAI2</td><td>DTSC7</td><td>Ubuntu V2</td><td>WikiArticleSearch</td></tr><tr><td>Train Ex.</td><td>131,438</td><td>100,000</td><td>1,000.000</td><td>5,035,182</td></tr><tr><td>Valid Ex.</td><td>7,801</td><td>10,000</td><td>19,560</td><td>9,921</td></tr><tr><td>Test Ex.</td><td>6634</td><td>5.000</td><td>18,920</td><td>9,925</td></tr><tr><td>Eval Cands per Ex.</td><td>20</td><td>100</td><td>10</td><td>10,001</td></tr></table>",
+        "bbox": [
+            246,
+            270,
+            746,
+            337
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 Methods ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            391,
+            276,
+            407
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section we describe the various models and methods that we explored. ",
+        "bbox": [
+            174,
+            424,
+            678,
+            439
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1 Transformers and Pre-training Strategies ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            458,
+            496,
+            472
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Transformers Our Bi-, Cross-, and Poly-encoders, described in sections 4.2, 4.3 and 4.4 respectively, are based on large pre-trained transformer models with the same architecture and dimension as BERT-base (Devlin et al., 2019), which has 12 layers, 12 attention heads, and a hidden size of 768. As well as considering the BERT pre-trained weights, we also explore our own pre-training schemes. Specifically, we pre-train two more transformers from scratch using the exact same architecture as BERT-base. One uses a similar training setup as in BERT-base, training on 150 million of examples of [INPUT, LABEL] extracted from Wikipedia and the Toronto Books Corpus, while the other is trained on 174 million examples of [INPUT, LABEL] extracted from the online platform Reddit (Mazare et al., 2018), which is a dataset more adapted to dialogue. The former is performed ´ to verify that reproducing a BERT-like setting gives us the same results as reported previously, while the latter tests whether pre-training on data more similar to the downstream tasks of interest helps. For training both new setups we used XLM (Lample & Conneau, 2019). ",
+        "bbox": [
+            174,
+            484,
+            825,
+            651
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Input Representation Our pre-training input is the concatenation of input and label [INPUT,LABEL], where both are surrounded with the special token [S], following Lample & Conneau (2019). When pre-training on Reddit, the input is the context, and the label is the next utterance. When pre-training on Wikipedia and Toronto Books, as in Devlin et al. (2019), the input is one sentence and the label the next sentence in the text. Each input token is represented as the sum of three embeddings: the token embedding, the position (in the sequence) embedding and the segment embedding. Segments for input tokens are 0, and for label tokens are 1. ",
+        "bbox": [
+            174,
+            670,
+            825,
+            767
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Pre-training Procedure Our pre-training strategy involves training with a masked language model (MLM) task identical to the one in Devlin et al. (2019). In the pre-training on Wikipedia and Toronto Books we add a next-sentence prediction task identical to BERT training. In the pre-training on Reddit, we add a next-utterance prediction task, which is slightly different from the previous one as an utterance can be composed of several sentences. During training $50 \\%$ of the time the candidate is the actual next sentence/utterance and $50 \\%$ of the time it is a sentence/utterance randomly taken from the dataset. We alternate between batches of the MLM task and the next-sentence/nextutterance prediction task. Like in Lample & Conneau (2019) we use the Adam optimizer with learning rate of 2e-4, $\\beta _ { 1 } = 0 . 9$ , $\\beta _ { 2 } = 0 . 9 8$ , no L2 weight decay, linear learning rate warmup, and β . β .inverse square root decay of the learning rate. We use a dropout probability of 0.1 on all layers, and a batch of 32000 tokens composed of concatenations [INPUT, LABEL] with similar lengths. We train the model on 32 GPUs for 14 days. ",
+        "bbox": [
+            174,
+            785,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Fine-tuning After pre-training, one can then fine-tune for the multi-sentence selection task of choice, in our case one of the four tasks from Section 3. We consider three architectures with which we fine-tune the transformer: the Bi-encoder, Cross-encoder and newly proposed Poly-encoder. ",
+        "bbox": [
+            174,
+            147,
+            825,
+            189
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 Bi-encoder ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            205,
+            285,
+            219
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In a Bi-encoder, both the input context and the candidate label are encoded into vectors: ",
+        "bbox": [
+            173,
+            231,
+            748,
+            246
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/88993570361f99fcb50fa44bac14ee4e8d67697b3e764d367a3251cd757ae38c.jpg",
+        "text": "$$\ny _ { c t x t } = r e d ( T _ { 1 } ( c t x t ) ) \\qquad y _ { c a n d } = r e d ( T _ { 2 } ( c a n d ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            252,
+            656,
+            267
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $T _ { 1 }$ and $T _ { 2 }$ are two transformers that have been pre-trained following the procedure described in 4.1; they initially start with the same weights, but are allowed to update separately during finetuning. $T ( x ) = h _ { 1 } , . . , h _ { N }$ is the output of a transformer $\\mathrm { T }$ and $r e d ( \\cdot )$ is a function that reduces that , ..,sequence of vectors into one vector. As the input and the label are encoded separately, segment tokens are 0 for both. To resemble what is done during our pre-training, both the input and label are surrounded by the special token [S] and therefore $h _ { 1 }$ corresponds to [S]. ",
+        "bbox": [
+            173,
+            273,
+            825,
+            358
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We considered three ways of reducing the output into one representation via red(·): choose the first output of the transformer (corresponding to the special token [S]), compute the average over all outputs or the average over the first $m \\leq N$ outputs. We compare them in Table 7 in the Appendix. We use the first output of the transformer in our experiments as it gives slightly better results. ",
+        "bbox": [
+            174,
+            364,
+            825,
+            421
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Scoring The score of a candidate candi is given by the dot-product $s ( c t x t , c a n d _ { i } ) = y _ { c t x t } \\cdot y _ { c a n d _ { i } } ,$ . The network is trained to minimize a cross-entropy loss in which the logits are $y _ { c t x t } \\cdot y _ { c a n d _ { 1 } } , . . . , y _ { c t x t } \\cdot y _ { c a n d _ { n } }$ , where cand $_ 1$ , ...,is the correct label and the others are chosen from the training set. Similar to Mazare´ et al. (2018), during training we consider the other labels in the batch as negatives. This allows for much faster training, as we can reuse the embeddings computed for each candidate, and also use a larger batch size; e.g., in our experiments on ConvAI2, we were able to use batches of 512 elements. ",
+        "bbox": [
+            174,
+            435,
+            825,
+            520
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Inference speed In the setting of retrieval over known candidates, a Bi-encoder allows for the precomputation of the embeddings of all possible candidates of the system. After the context embedding $y _ { c t x t }$ is computed, the only operation remaining is a dot product between $y _ { c t x t }$ and every candidate embedding, which can scale to millions of candidates on a modern GPU, and potentially billions using nearest-neighbor libraries such as FAISS (Johnson et al., 2019). ",
+        "bbox": [
+            174,
+            534,
+            825,
+            604
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.3 Cross-encoder ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            621,
+            310,
+            635
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Cross-encoder allows for rich interactions between the input context and candidate label, as they are jointly encoded to obtain a final representation. Similar to the procedure in pre-training, the context and candidate are surrounded by the special token [S] and concatenated into a single vector, which is encoded using one transformer. We consider the first output of the transformer as the context-candidate embedding: ",
+        "bbox": [
+            174,
+            646,
+            825,
+            717
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/896d60637b8df66570153eb2312c951bd471ee7611c9802dea1f3d62fe6d2dd2.jpg",
+        "text": "$$\ny _ { c t x t , c a n d } = h _ { 1 } = f i r s t ( T ( c t x t , c a n d ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            722,
+            617,
+            738
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where f irst is the function that takes the first vector of the sequence of vectors produced by the transformer. By using a single transformer, the Cross-encoder is able to perform self-attention between the context and candidate, resulting in a richer extraction mechanism than the Bi-encoder. As the candidate label can attend to the input context during the layers of the transformer, the Crossencoder can produce a candidate-sensitive input representation, which the Bi-encoder cannot. For example, this allows it to select useful input features per candidate. ",
+        "bbox": [
+            174,
+            743,
+            825,
+            828
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Scoring To score one candidate, a linear layer $W$ is applied to the embedding $y _ { c t x t , c a n d }$ to reduce it from a vector to a scalar: ",
+        "bbox": [
+            173,
+            842,
+            823,
+            869
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/286ebe0a65fa7804fa42a9e3fa007b3b1d4fb85a15a0c3fde8ba1b1ca9071452.jpg",
+        "text": "$$\ns ( c t x t , c a n d _ { i } ) = y _ { c t x t , c a n d _ { i } } W\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            869,
+            588,
+            886
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similarly to what is done for the Bi-encoder, the network is trained to minimize a cross entropy loss where the logits are $s ( c t x t , c a n d _ { 1 } ) , . . . , s ( c t x t , c a n d _ { n } )$ , where can $l _ { 1 }$ is the correct candidate and the others are negatives taken from the training set. Unlike in the Bi-encoder, we cannot recycle the other labels of the batch as negatives, so we use external negatives provided in the training set. The Cross-encoder uses much more memory than the Bi-encoder, resulting in a much smaller batch size. ",
+        "bbox": [
+            174,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/3110c1d61ef5e5558af249f14c69c29298a158c4e446683e10affa39d7c316ba.jpg",
+        "image_caption": [
+            "Figure 1: Diagrams of the three model architectures we consider. (a) The Bi-encoder encodes the context and candidate separately, allowing for the caching of candidate representations during inference. (b) The Cross-encoder jointly encodes the context and candidate in a single transformer, yielding richer interactions between context and candidate at the cost of slower computation. (c) The Poly-encoder combines the strengths of the Bi-encoder and Cross-encoder by both allowing for caching of candidate representations and adding a final attention mechanism between global features of the input and a given candidate to give richer interactions before computing a final score. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            101,
+            759,
+            439
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            571,
+            825,
+            614
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Inference speed Unfortunately, the Cross-encoder does not allow for precomputation of the candidate embeddings. At inference time, every candidate must be concatenated with the input context and must go through a forward pass of the entire model. Thus, this method cannot scale to a large amount of candidates. We discuss this bottleneck further in Section 5.4. ",
+        "bbox": [
+            174,
+            630,
+            825,
+            685
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.4 Poly-encoder ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            703,
+            299,
+            717
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The Poly-encoder architecture aims to get the best of both worlds from the Bi- and Cross-encoder. A given candidate label is represented by one vector as in the Bi-encoder, which allows for caching candidates for fast inference time, while the input context is jointly encoded with the candidate, as in the Cross-encoder, allowing the extraction of more information. ",
+        "bbox": [
+            174,
+            728,
+            825,
+            784
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The Poly-encoder uses two separate transformers for the context and label like a Bi-encoder, and the candidate is encoded into a single vector $y _ { c a n d _ { i } }$ . As such, the Poly-encoder method can be implemented using a precomputed cache of encoded responses. However, the input context, which is typically much longer than a candidate, is represented with $m$ vectors $( y _ { c t x t } ^ { 1 } . . . y _ { c t x t } ^ { m } )$ instead of just one as in the Bi-encoder, where $m$ ..will influence the inference speed. To obtain these $m$ global features that represent the input, we learn $m$ context codes $( c _ { 1 } , . . . , c _ { m } )$ , where $c _ { i }$ extracts representation $y _ { c t x t } ^ { i }$ , ...,by attending over all the outputs of the previous layer. That is, we obtain $y _ { c t x t } ^ { i }$ using: ",
+        "bbox": [
+            174,
+            791,
+            825,
+            890
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1344da801c8e5645b24b45749984ba65c2620f65fc7723a119e4028fe14d6b59.jpg",
+        "text": "$$\ny _ { c t x t } ^ { i } = \\sum _ { j } w _ { j } ^ { c _ { i } } h _ { j } ~ \\mathrm { w h e r e } ~ ( w _ { 1 } ^ { c _ { i } } , . . , w _ { N } ^ { c _ { i } } ) = \\mathrm { s o f t m a x } ( c _ { i } \\cdot h _ { 1 } , . . , c _ { i } \\cdot h _ { N } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            892,
+            727,
+            928
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The $m$ context codes are randomly initialized, and learnt during finetuning. ",
+        "bbox": [
+            174,
+            103,
+            663,
+            118
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, given our $m$ global context features, we attend over them using $y _ { c a n d _ { i } }$ as the query: ",
+        "bbox": [
+            166,
+            125,
+            761,
+            140
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1681f6147cdaeddac93163092077f471a6647120634381f0beea7cf580612ee8.jpg",
+        "text": "$$\ny _ { c t x t } = \\sum _ { i } w _ { i } y _ { c t x t } ^ { i } ~ \\mathrm { w h e r e } ~ ( w _ { 1 } , . . , w _ { m } ) = \\mathrm { s o f t m a x } ( y _ { c a n d _ { i } } \\cdot y _ { c t x t } ^ { 1 } , . . , y _ { c a n d _ { i } } \\cdot y _ { c t x t } ^ { m } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            145,
+            761,
+            179
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The final score for that candidate label is then $y _ { c t x t } \\cdot y _ { c a n d _ { i } }$ as in a Bi-encoder. As $m < N$ , where $N$ is <the number of tokens, and the context-candidate attention is only performed at the top layer, this is far faster than the Cross-encoder’s full self-attention. ",
+        "bbox": [
+            174,
+            185,
+            821,
+            228
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            247,
+            303,
+            263
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We perform a variety of experiments to test our model architectures and training strategies over four tasks. For metrics, we measure Recall $@ k$ where each test example has $C$ possible candidates to select from, abbreviated to $\\operatorname { R @ } k / C$ , as well as mean reciprocal rank (MRR). ",
+        "bbox": [
+            176,
+            279,
+            825,
+            321
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 Bi-encoders and Cross-encoders ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            338,
+            428,
+            352
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We first investigate fine-tuning the Bi- and Cross-encoder architectures initialized with the weights provided by Devlin et al. (2019), studying the choice of other hyperparameters (we explore our own pre-training schemes in section 5.3). In the case of the Bi-encoder, we can use a large number of negatives by considering the other batch elements as negative training samples, avoiding recomputation of their embeddings. On 8 Nvidia Volta v100 GPUs and using half-precision operations (i.e. float16 operations), we can reach batches of 512 elements on ConvAI2. Table 2 shows that in this setting, we obtain higher performance with a larger batch size, i.e. more negatives, where 511 negatives yields the best results. For the other tasks, we keep the batch size at 256, as the longer sequences in those datasets uses more memory. The Cross-encoder is more computationally intensive, as the embeddings for the (context, candidate) pair must be recomputed each time. We thus limit its batch size to 16 and provide negatives random samples from the training set. For DSTC7 and Ubuntu V2, we choose 15 such negatives; For ConvAI2, the dataset provides 19 negatives. ",
+        "bbox": [
+            174,
+            363,
+            825,
+            531
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/91e2aa52e8d79bad4c869865e1eef83cc6f2abe7c79ced063eae4761686d933f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Negatives</td><td rowspan=1 colspan=1>31</td><td rowspan=1 colspan=1>63</td><td rowspan=1 colspan=1>127</td><td rowspan=1 colspan=1>255</td><td rowspan=1 colspan=1>511</td></tr><tr><td rowspan=1 colspan=1>R@1/20</td><td rowspan=1 colspan=1>81.0</td><td rowspan=1 colspan=1>81.7</td><td rowspan=1 colspan=1>82.3</td><td rowspan=1 colspan=1>83.0</td><td rowspan=1 colspan=1>83.3</td></tr></table>",
+        "bbox": [
+            333,
+            544,
+            665,
+            575
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Validation performance on ConvAI2 after fine-tuning a Bi-encoder pre-trained with BERT, averaged over 5 runs. The batch size is the number of training negatives $^ { + 1 }$ as we use the other elements of the batch as negatives during training. ",
+        "bbox": [
+            174,
+            585,
+            825,
+            628
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The above results are reported with Bi-encoder aggregation based on the first output. Choosing the average over all outputs instead is very similar but slightly worse (83.1, averaged over 5 runs). We also tried to add further non-linearities instead of the inner product of the two representations, but could not obtain improved results over the simpler architecture (results not shown). ",
+        "bbox": [
+            174,
+            645,
+            825,
+            702
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We tried two optimizers: Adam (Kingma & Ba, 2015) with weight decay of 0.01 (as recommended by (Devlin et al., 2019)) and Adamax (Kingma & Ba, 2015) without weight decay; based on validation set performance, we choose to fine-tune with Adam when using the BERT weights. The learning rate is initialized to 5e-5 with a warmup of 100 iterations for Bi- and Poly-encoders, and 1000 iterations for the Cross-encoder. The learning rate decays by a factor of 0.4 upon plateau of the loss evaluated on the valid set every half epoch. In Table 3 we show validation performance when fine-tuning various layers of the weights provided by (Devlin et al., 2019), using Adam with decay optimizer. Fine-tuning the entire network is important, with the exception of the word embeddings. ",
+        "bbox": [
+            174,
+            707,
+            825,
+            819
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "With the setups described above, we fine-tune the Bi- and Cross-encoders on the datasets, and report the results in Table 4. On the first three tasks, our Bi-encoders and Cross-encoders outperform the best existing approaches in the literature when we fine-tune from BERT weights. E.g., the Biencoder reaches $8 1 . 7 \\%$ $\\mathbb { R } \\ @ 1$ on ConvAI2 and $6 6 . 8 \\%$ $\\mathbb { R } \\ @ 1$ on DSTC7, while the Cross-encoder achieves higher scores of $8 4 . 8 \\%$ $\\mathbb { R } \\ @ 1$ on ConvAI2 and $6 7 . 4 \\%$ $\\mathbf { R } \\ @ 1$ on DSTC7. Overall, Crossencoders outperform all previous approaches on the three dialogue tasks, including our Bi-encoders (as expected). We do not report fine-tuning of BERT for Wikipedia IR as we cannot guarantee the test set is not part of the pre-training for that dataset. In addition, Cross-encoders are also too slow to evaluate on the evaluation setup of that task, which has 10k candidates. ",
+        "bbox": [
+            174,
+            825,
+            823,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a6f8fe18e6dadc1ad50ee675d0713548b36393e622c6e68bf279c848519d1d0.jpg",
+        "table_caption": [
+            "Table 3: Validation performance $( \\mathbf { R } @ 1 / 2 0 )$ on ConvAI2 using pre-trained weights of BERT-base with different parameters fine-tuned. Average over 5 runs (Bi-encoders) or 3 runs (Cross-encoders). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Fine-tuned parameters</td><td>Bi-encoder</td><td>Cross-encoder</td></tr><tr><td>Top layer</td><td>74.2</td><td>80.6</td></tr><tr><td>Top 4 layers</td><td>82.0</td><td>86.3</td></tr><tr><td>All but Embeddings</td><td>83.3</td><td>87.3</td></tr><tr><td>Every Layer</td><td>83.0</td><td>86.6</td></tr></table>",
+        "bbox": [
+            325,
+            101,
+            673,
+            174
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            237,
+            823,
+            266
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/f2bb9160a711d284c0035ea3e21d60d0eff122abc814395c223f1ec69283d35b.jpg",
+        "table_caption": [
+            "Table 4: Test performance of Bi-, Poly- and Cross-encoders on our selected tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>ConvAI2</td><td rowspan=1 colspan=2>DSTC7</td><td rowspan=1 colspan=2>Ubuntu v2</td><td rowspan=1 colspan=1>Wikipedia IR</td></tr><tr><td rowspan=1 colspan=1>split</td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=2>test</td><td rowspan=1 colspan=2>test</td><td rowspan=1 colspan=1>test</td></tr><tr><td rowspan=1 colspan=1>metric</td><td rowspan=1 colspan=1>R@1/20</td><td rowspan=1 colspan=1>R@1/100</td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>R@1/10</td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>R@1/10001</td></tr><tr><td rowspan=1 colspan=1>(Wolf et al., 2019)</td><td rowspan=1 colspan=1>80.7</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>(Gu et al., 2018)</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>60.8</td><td rowspan=1 colspan=1>69.1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Chen &amp;Wang,2019)</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>64.5</td><td rowspan=1 colspan=1>73.5</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Yoon et al.,2018)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>65.2</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Dong&amp; Huang,2018)</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>75.9</td><td rowspan=1 colspan=1>84.8</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Wu et al.,2018)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>56.8</td></tr><tr><td rowspan=1 colspan=1>pre-trainedBERTweigh</td><td rowspan=1 colspan=1>tsfrom (De</td><td rowspan=1 colspan=1>linetal.,20</td><td rowspan=1 colspan=1>19)-Toron</td><td rowspan=1 colspan=1>oBooks+</td><td rowspan=1 colspan=1>Vikipedia</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>81.7 ± 0.2</td><td rowspan=1 colspan=1>66.8 ± 0.7</td><td rowspan=1 colspan=1>74.6 ± 0.5</td><td rowspan=1 colspan=1>80.6 ± 0.4</td><td rowspan=1 colspan=1>88.0±0.3</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder16</td><td rowspan=1 colspan=1>83.2 ± 0.1</td><td rowspan=1 colspan=1>67.8 ± 0.3</td><td rowspan=1 colspan=1>75.1 ± 0.2</td><td rowspan=1 colspan=1>81.2 ± 0.2</td><td rowspan=1 colspan=1>88.3± 0.1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 64</td><td rowspan=1 colspan=1>83.7 ± 0.2</td><td rowspan=1 colspan=1>67.0 ± 0.9</td><td rowspan=1 colspan=1>74.7 ± 0.6</td><td rowspan=1 colspan=1>81.3 ± 0.2</td><td rowspan=1 colspan=1>88.4 ± 0.1</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 360</td><td rowspan=1 colspan=1>83.7 ± 0.2</td><td rowspan=1 colspan=1>68.9± 0.4</td><td rowspan=1 colspan=1>76.2 ± 0.2</td><td rowspan=1 colspan=1>80.9± 0.0</td><td rowspan=1 colspan=1>88.1 ± 0.1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>84.8 ± 0.3</td><td rowspan=1 colspan=1>67.4 ± 0.7</td><td rowspan=1 colspan=1>75.6 ± 0.4</td><td rowspan=1 colspan=1>82.8 ± 0.3</td><td rowspan=1 colspan=1>89.4 ± 0.2</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Our pre-training on Toronto Books + </td><td rowspan=1 colspan=1>ntoBooks-</td><td rowspan=1 colspan=1>Wikipedia</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>82.0 ± 0.1</td><td rowspan=1 colspan=1>64.5 ± 0.5</td><td rowspan=1 colspan=1>72.6 ± 0.4</td><td rowspan=1 colspan=1>80.8± 0.5</td><td rowspan=1 colspan=1>88.2 ± 0.4</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 16</td><td rowspan=1 colspan=1>82.7 ± 0.1</td><td rowspan=1 colspan=1>65.3 ± 0.9</td><td rowspan=1 colspan=1>73.2 ± 0.7</td><td rowspan=1 colspan=1>83.4± 0.2</td><td rowspan=1 colspan=1>89.9 ± 0.1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 64</td><td rowspan=1 colspan=1>83.3 ± 0.1</td><td rowspan=1 colspan=1>65.8 ± 0.7</td><td rowspan=1 colspan=1>73.5 ± 0.5</td><td rowspan=1 colspan=1>83.4± 0.1</td><td rowspan=1 colspan=1>89.9± 0.0</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 360</td><td rowspan=1 colspan=1>83.8 ± 0.1</td><td rowspan=1 colspan=1>65.8 ± 0.7</td><td rowspan=1 colspan=1>73.6 ± 0.6</td><td rowspan=1 colspan=1>83.7±0.0</td><td rowspan=1 colspan=1>90.1 ± 0.0</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>84.9 ± 0.3</td><td rowspan=1 colspan=1>65.3 ± 1.0</td><td rowspan=1 colspan=1>73.8± 0.6</td><td rowspan=1 colspan=1>83.1 ± 0.7</td><td rowspan=1 colspan=1>89.7 ± 0.5</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Our pre-training on Reddit</td><td rowspan=1 colspan=1>dit</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>84.8± 0.1</td><td rowspan=1 colspan=1>70.9 ± 0.5</td><td rowspan=1 colspan=1>78.1 ± 0.3</td><td rowspan=1 colspan=1>83.6± 0.7</td><td rowspan=1 colspan=1>90.1 ± 0.4</td><td rowspan=1 colspan=1>71.0</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 16</td><td rowspan=1 colspan=1>86.3 ± 0.3</td><td rowspan=1 colspan=1>71.6 ± 0.6</td><td rowspan=1 colspan=1>78.4± 0.4</td><td rowspan=1 colspan=1>86.0± 0.1</td><td rowspan=1 colspan=1>91.5 ± 0.1</td><td rowspan=1 colspan=1>71.5</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 64</td><td rowspan=1 colspan=1>86.5± 0.2</td><td rowspan=1 colspan=1>71.2 ± 0.8</td><td rowspan=1 colspan=1>78.2 ± 0.7</td><td rowspan=1 colspan=1>85.9± 0.1</td><td rowspan=1 colspan=1>91.5 ± 0.1</td><td rowspan=1 colspan=1>71.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 360</td><td rowspan=1 colspan=1>86.8 ± 0.1</td><td rowspan=1 colspan=1>71.4 ± 1.0</td><td rowspan=1 colspan=1>78.3 ± 0.7</td><td rowspan=1 colspan=1>85.9 ± 0.1</td><td rowspan=1 colspan=1>91.5 ± 0.0</td><td rowspan=1 colspan=1>71.8</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>87.9 ± 0.2</td><td rowspan=1 colspan=1>71.7 ± 0.3</td><td rowspan=1 colspan=1>79.0 ± 0.2</td><td rowspan=1 colspan=1>86.5 ± 0.1</td><td rowspan=1 colspan=1>91.9 ± 0.0</td><td rowspan=1 colspan=1>-</td></tr></table>",
+        "bbox": [
+            197,
+            290,
+            799,
+            659
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 Poly-encoders ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            724,
+            307,
+            739
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We train the Poly-encoder using the same batch sizes and optimizer choices as in the Bi-encoder experiments. Results are reported in Table 4 for various values of $m$ context vectors. ",
+        "bbox": [
+            174,
+            756,
+            823,
+            784
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The Poly-encoder outperforms the Bi-encoder on all the tasks, with more codes generally yielding larger improvements. Our recommendation is thus to use as large a code size as compute time allows (see Sec. 5.4). On DSTC7, the Poly-encoder architecture with BERT pretraining reaches $6 8 . 9 \\%$ R1 with 360 intermediate context codes; this actually outperforms the Cross-encoder result $( 6 7 . 4 \\% )$ and is noticeably better than our Bi-encoder result $( 6 6 . 8 \\% )$ . Similar conclusions are found on Ubuntu V2 and ConvAI2, although in the latter Cross-encoders give slightly better results. ",
+        "bbox": [
+            174,
+            790,
+            825,
+            875
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We note that since reporting our results, the authors of Li et al. (2019) have conducted a human evaluation study on ConvAI2, in which our Poly-encoder architecture outperformed all other models compared against, both generative and retrieval based, including the winners of the competition. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/332334339c64f6ba612a777428a7b398b91b5e4274620fe45030c80d604a5c52.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=4>Scoring time (ms)</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>CPU</td><td rowspan=1 colspan=2>GPU</td></tr><tr><td rowspan=1 colspan=1>Candidates</td><td rowspan=1 colspan=1>1k</td><td rowspan=1 colspan=1>100k</td><td rowspan=1 colspan=1>1k</td><td rowspan=1 colspan=1>100k</td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>115</td><td rowspan=1 colspan=1>160</td><td rowspan=1 colspan=1>19</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 16</td><td rowspan=1 colspan=1>122</td><td rowspan=1 colspan=1>678</td><td rowspan=1 colspan=1>18</td><td rowspan=1 colspan=1>38</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 64</td><td rowspan=1 colspan=1>126</td><td rowspan=1 colspan=1>692</td><td rowspan=1 colspan=1>23</td><td rowspan=1 colspan=1>46</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 360</td><td rowspan=1 colspan=1>160</td><td rowspan=1 colspan=1>837</td><td rowspan=1 colspan=1>57</td><td rowspan=1 colspan=1>88</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>21.7k</td><td rowspan=1 colspan=1>2.2M*</td><td rowspan=1 colspan=1>2.6k</td><td rowspan=1 colspan=1>266k*</td></tr></table>",
+        "bbox": [
+            330,
+            101,
+            665,
+            222
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Average time in milliseconds to predict the next dialogue utterance from $C$ possible candidates on ConvAI2. \\* are inferred. ",
+        "bbox": [
+            169,
+            232,
+            821,
+            260
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 Domain-specific Pre-training ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            296,
+            405,
+            310
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We fine-tune our Reddit-pre-trained transformer on all four tasks; we additionally fine-tune a transformer that was pre-trained on the same datasets as BERT, specifically Toronto Books $^ +$ Wikipedia. When using our pre-trained weights, we use the Adamax optimizer and optimize all the layers of the transformer including the embeddings. As we do not use weight decay, the weights of the final layer are much larger than those in the final layer of BERT; to avoid saturation of the attention layer in the Poly-encoder, we re-scaled the last linear layer so that the standard deviation of its output matched that of BERT, which we found necessary to achieve good results. We report results of fine-tuning with our pre-trained weights in Table 4. We show that pre-training on Reddit gives further state-ofthe-art performance over our previous results with BERT, a finding that we see for all three dialogue tasks, and all three architectures. ",
+        "bbox": [
+            174,
+            333,
+            825,
+            472
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The results obtained with fine-tuning on our own transformers pre-trained on Toronto Books $^ +$ Wikipedia are very similar to those obtained with the original BERT weights, indicating that the choice of dataset used to pre-train the models impacts the final results, not some other detail in our training. Indeed, as the two settings pre-train with datasets of similar size, we can conclude that choosing a pre-training task (e.g. dialogue data) that is similar to the downstream tasks of interest (e.g. dialogue) is a likely explanation for these performance gains, in line with previous results showing multi-tasking with similar tasks is more useful than with dissimilar ones (Caruana, 1997). ",
+        "bbox": [
+            174,
+            479,
+            825,
+            577
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 Inference Speed ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            622,
+            318,
+            636
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "An important motivation for the Poly-encoder architecture is to achieve better results than the Biencoder while also performing at a reasonable speed. Though the Cross-encoder generally yields strong results, it is prohibitively slow. We perform speed experiments to determine the trade-off of improved performance from the Poly-encoder. Specifically, we predict the next utterance for 100 dialogue examples in the ConvAI2 validation set, where the model scores $C$ candidates (in this case, chosen from the training set). We perform these experiments on both CPU-only and GPU setups. CPU computations were run on an 80 core Intel Xeon processor CPU E5-2698. GPU computations were run on a single Nvidia Quadro GP100 using cuda 10.0 and cudnn 7.4. ",
+        "bbox": [
+            174,
+            659,
+            825,
+            770
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We show the average time per example for each architecture in Table 5. The difference in timing between the Bi-encoder and the Poly-encoder architectures is rather minimal when there are only 1000 candidates for the model to consider. The difference is more pronounced when considering 100k candidates, a more realistic setup, as we see a 5-6x slowdown for the Poly-encoder variants. Nevertheless, both models are still tractable. The Cross-encoder, however, is 2 orders of magnitude slower than the Bi-encoder and Poly-encoder, rendering it intractable for real-time inference, e.g. when interacting with a dialogue agent, or retrieving from a large set of documents. Thus, Polyencoders, given their desirable performance and speed trade-off, are the preferred method. ",
+        "bbox": [
+            174,
+            776,
+            825,
+            888
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We additionally report training times in the Appendix, Table 6. Poly-encoders also have the benefit of being $3 { - } 4 \\mathbf { x }$ faster to train than Cross-encoders (and are similar in training time to Bi-encoders). ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            295,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper we present new architectures and pre-training strategies for deep bidirectional transformers in candidate selection tasks. We introduced the Poly-encoder method, which provides a mechanism for attending over the context using the label candidate, while maintaining the ability to precompute each candidate’s representation, which allows for fast real-time inference in a production setup, giving an improved trade off between accuracy and speed. We provided an experimental analysis of those trade-offs for Bi-, Poly- and Cross-encoders, showing that Poly-encoders are more accurate than Bi-encoders, while being far faster than Cross-encoders, which are impractical for real-time use. In terms of training these architectures, we showed that pre-training strategies more closely related to the downstream task bring strong improvements. In particular, pre-training from scratch on Reddit allows us to outperform the results we obtain with BERT, a result that holds for all three model architectures and all three dialogue datasets we tried. However, the methods introduced in this work are not specific to dialogue, and can be used for any task where one is scoring a set of candidates, which we showed for an information retrieval task as well. ",
+        "bbox": [
+            174,
+            135,
+            825,
+            314
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            337,
+            264,
+            353
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Bing Bai, Jason Weston, David Grangier, Ronan Collobert, Kunihiko Sadamasa, Yanjun Qi, Olivier Chapelle, and Kilian Weinberger. Supervised semantic indexing. In Proceedings of the 18th ACM conference on Information and knowledge management, pp. 187–196. ACM, 2009. ",
+        "bbox": [
+            174,
+            362,
+            825,
+            405
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jane Bromley, Isabelle Guyon, Yann LeCun, Eduard Sackinger, and Roopak Shah. Signature verifi- ¨ cation using a” siamese” time delay neural network. In Advances in neural information processing systems, pp. 737–744, 1994. ",
+        "bbox": [
+            174,
+            416,
+            823,
+            458
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Rich Caruana. Multitask learning. Machine learning, 28(1):41–75, 1997. ",
+        "bbox": [
+            173,
+            470,
+            653,
+            486
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Qian Chen and Wen Wang. Sequential attention-based network for noetic end-to-end response selection. CoRR, abs/1901.02609, 2019. URL http://arxiv.org/abs/1901.02609. ",
+        "bbox": [
+            174,
+            498,
+            823,
+            526
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lazaros Polymenakos Chulaka Gunasekara, Jonathan K. Kummerfeld and Walter S. Lasecki. Dstc7 task 1: Noetic end-to-end response selection. In 7th Edition of the Dialog System Technology Challenges at AAAI 2019, January 2019. URL http://workshop.colips.org/dstc7/ papers/dstc7_task1_final_report.pdf. ",
+        "bbox": [
+            174,
+            537,
+            825,
+            594
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Scott Deerwester, Susan T Dumais, George W Furnas, Thomas K Landauer, and Richard Harshman. Indexing by latent semantic analysis. Journal of the American society for information science, 41 (6):391–407, 1990. ",
+        "bbox": [
+            174,
+            608,
+            823,
+            650
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pp. 4171–4186, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics. doi: 10.18653/v1/N19-1423. ",
+        "bbox": [
+            173,
+            661,
+            825,
+            732
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Emily Dinan, Stephen Roller, Kurt Shuster, Angela Fan, Michael Auli, and Jason Weston. Wizard of Wikipedia: Knowledge-powered conversational agents. In Proceedings of the International Conference on Learning Representations (ICLR), 2019. ",
+        "bbox": [
+            174,
+            744,
+            823,
+            786
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Emily Dinan, Varvara Logacheva, Valentin Malykh, Alexander Miller, Kurt Shuster, Jack Urbanek, Douwe Kiela, Arthur Szlam, Iulian Serban, Ryan Lowe, Shrimai Prabhumoye, Alan W. Black, Alexander Rudnicky, Jason Williams, Joelle Pineau, Mikhail Burtsev, and Jason Weston. The second conversational intelligence challenge (convai2). In Sergio Escalera and Ralf Herbrich (eds.), The NeurIPS ’18 Competition, pp. 187–208, Cham, 2020. Springer International Publishing. ISBN 978-3-030-29135-8. ",
+        "bbox": [
+            174,
+            799,
+            825,
+            882
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jianxiong Dong and Jim Huang. Enhance word representation for out-of-vocabulary on ubuntu dialogue corpus. CoRR, abs/1802.02614, 2018. URL http://arxiv.org/abs/1802.02614. ",
+        "bbox": [
+            176,
+            896,
+            821,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jia-Chen Gu, Zhen-Hua Ling, Yu-Ping Ruan, and Quan Liu. Building sequential inference models for end-to-end response selection. CoRR, abs/1812.00686, 2018. URL http://arxiv.org/ abs/1812.00686. ",
+        "bbox": [
+            173,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "J. Johnson, M. Douze, and H. Jgou. Billion-scale similarity search with gpus. IEEE Transactions on Big Data, pp. 1–1, 2019. ISSN 2372-2096. doi: 10.1109/TBDATA.2019.2921572. ",
+        "bbox": [
+            174,
+            157,
+            821,
+            186
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Joseph Peper Vignesh Athreya Chulaka Gunasekara Jatin Ganhotra Siva Sankalp Patel Lazaros Polymenakos Jonathan K. Kummerfeld, Sai R. Gouravajhala and Walter S. Lasecki. Analyzing assumptions in conversation disentanglement research through the lens of a new dataset and model. ArXiv e-prints, October 2018. URL https://arxiv.org/pdf/1810.11118.pdf. ",
+        "bbox": [
+            173,
+            199,
+            825,
+            256
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Rudolf Kadlec, Martin Schmid, and Jan Kleindienst. Improved deep learning baselines for ubuntu corpus dialogs. CoRR, abs/1510.03753, 2015. URL http://arxiv.org/abs/1510.03753. ",
+        "bbox": [
+            169,
+            267,
+            825,
+            296
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015. URL http://arxiv.org/abs/1412.6980. ",
+        "bbox": [
+            173,
+            308,
+            825,
+            351
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Guillaume Lample and Alexis Conneau. Cross-lingual language model pretraining. Advances in Neural Information Processing Systems (NeurIPS), 2019. ",
+        "bbox": [
+            174,
+            363,
+            823,
+            392
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Margaret Li, Jason Weston, and Stephen Roller. Acute-eval: Improved dialogue evaluation with optimized questions and multi-turn comparisons. arXiv preprint arXiv:1909.03087, 2019. ",
+        "bbox": [
+            171,
+            404,
+            823,
+            433
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ryan Lowe, Nissan Pow, Iulian Serban, and Joelle Pineau. The ubuntu dialogue corpus: A large dataset for research in unstructured multi-turn dialogue systems. In SIGDIAL Conference, 2015. ",
+        "bbox": [
+            173,
+            444,
+            823,
+            473
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Pierre-Emmanuel Mazare, Samuel Humeau, Martin Raison, and Antoine Bordes. Training millions ´ of personalized dialogue agents. In EMNLP, 2018. ",
+        "bbox": [
+            173,
+            484,
+            823,
+            513
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gerard Salton, Anita Wong, and Chung-Shu Yang. A vector space model for automatic indexing. Communications of the ACM, 18(11):613–620, 1975. ",
+        "bbox": [
+            171,
+            526,
+            823,
+            554
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jack Urbanek, Angela Fan, Siddharth Karamcheti, Saachi Jain, Samuel Humeau, Emily Dinan, Tim Rocktaschel, Douwe Kiela, Arthur Szlam, and Jason Weston. Learning to speak and act in a ¨ fantasy text adventure game. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pp. 673–683, Hong Kong, China, November 2019. Association for Computational Linguistics. doi: 10.18653/v1/D19-1062. ",
+        "bbox": [
+            173,
+            565,
+            825,
+            650
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jesse Vig and Kalai Ramea. Comparison of transfer-learning approaches for response selection in multi-turn conversations. Workshop on DSTC7, 2019. ",
+        "bbox": [
+            171,
+            662,
+            823,
+            691
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Thomas Wolf, Victor Sanh, Julien Chaumond, and Clement Delangue. Transfertransfo: A transfer learning approach for neural network based conversational agents. arXiv preprint arXiv:1901.08149, 2019. ",
+        "bbox": [
+            174,
+            704,
+            823,
+            746
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ledell Yu Wu, Adam Fisch, Sumit Chopra, Keith Adams, Antoine Bordes, and Jason Weston. Starspace: Embed all the things! In Thirty-Second AAAI Conference on Artificial Intelligence, 2018. ",
+        "bbox": [
+            173,
+            758,
+            825,
+            800
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yu Ping Wu, Wei Chung Wu, Chen Xing, Ming Zhou, and Zhoujun Li. Sequential matching network: A new architecture for multi-turn response selection in retrieval-based chatbots. In ACL, 2017. ",
+        "bbox": [
+            174,
+            813,
+            825,
+            854
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Liu Yang, Minghui Qiu, Chen Qu, Jiafeng Guo, Yongfeng Zhang, W Bruce Croft, Jun Huang, and Haiqing Chen. Response ranking with deep matching networks and external knowledge in information-seeking conversation systems. In The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval, pp. 245–254. ACM, 2018. ",
+        "bbox": [
+            173,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Seunghyun Yoon, Joongbo Shin, and Kyomin Jung. Learning to rank question-answer pairs using hierarchical recurrent encoder with latent topic clustering. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), pp. 1575–1584, New Orleans, Louisiana, June 2018. Association for Computational Linguistics. doi: 10.18653/v1/N18-1142. ",
+        "bbox": [
+            176,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Koichiro Yoshino, Chiori Hori, Julien Perez, Luis Fernando D’Haro, Lazaros Polymenakos, R. Chulaka Gunasekara, Walter S. Lasecki, Jonathan K. Kummerfeld, Michel Galley, Chris Brockett, Jianfeng Gao, William B. Dolan, Xiang Gao, Huda AlAmri, Tim K. Marks, Devi Parikh, and Dhruv Batra. Dialog system technology challenge 7. CoRR, abs/1901.03461, 2019. ",
+        "bbox": [
+            174,
+            183,
+            825,
+            239
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Saizheng Zhang, Emily Dinan, Jack Urbanek, Arthur Szlam, Douwe Kiela, and Jason Weston. Personalizing dialogue agents: I have a dog, do you have pets too? In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics, pp. 2204–2213, Melbourne, Australia, July 2018a. Association for Computational Linguistics. ",
+        "bbox": [
+            173,
+            247,
+            825,
+            304
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhuosheng Zhang, Jiangtong Li, Pengfei Zhu, Hai Zhao, and Gongshen Liu. Modeling multi-turn conversation with deep utterance aggregation. In COLING, 2018b. ",
+        "bbox": [
+            173,
+            313,
+            823,
+            342
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yukun Zhu, Ryan Kiros, Richard S. Zemel, Ruslan R. Salakhutdinov, Raquel Urtasun, Antonio Torralba, and Sanja Fidler. Aligning books and movies: Towards story-like visual explanations by watching movies and reading books. 2015 IEEE International Conference on Computer Vision (ICCV), pp. 19–27, 2015. ",
+        "bbox": [
+            174,
+            351,
+            825,
+            407
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A Training Time ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            321,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We report the training time on 8 GPU Volta 100 for the 3 datasets considered and for 4 types of models in Table 6. ",
+        "bbox": [
+            173,
+            133,
+            825,
+            161
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/95675a2b52a73fe0b3a87895b8f2228cfe9065bfcc7d593a8fbcad45b4ba70d6.jpg",
+        "table_caption": [
+            "Table 6: Training time in hours. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>ConvAI2</td><td rowspan=1 colspan=1>DSTC7</td><td rowspan=1 colspan=1>UbuntuV2</td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>2.0</td><td rowspan=1 colspan=1>4.9</td><td rowspan=1 colspan=1>7.9</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 16</td><td rowspan=1 colspan=1>2.7</td><td rowspan=1 colspan=1>5.5</td><td rowspan=1 colspan=1>8.0</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder 64</td><td rowspan=1 colspan=1>2.8</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>8.0</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder64</td><td rowspan=1 colspan=1>9.4</td><td rowspan=1 colspan=1>13.5</td><td rowspan=1 colspan=1>39.9</td></tr></table>",
+        "bbox": [
+            325,
+            172,
+            668,
+            241
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B Reduction layer in Bi-encoder ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            297,
+            450,
+            313
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We provide in Table 7 the results obtained for different types of reductions on top of the Bi-encoder. Specifically we compare the Recall $@$ 1/20 on the ConvAI2 validation set when taking the first output of BERT, the average of the first 16 outputs, the average of the first 64 outputs and all of them except the first one ([S]). ",
+        "bbox": [
+            174,
+            328,
+            826,
+            385
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/70a3643c57a6989a04f76d24ea3ac14d1e4a881715f34f8a1fe6ed1b7b2d792e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Setup</td><td rowspan=1 colspan=1>ConvAI2 valid Recall@1/20</td></tr><tr><td rowspan=1 colspan=1>First output</td><td rowspan=1 colspan=1>83.3</td></tr><tr><td rowspan=1 colspan=1>Avg first 16 outputs</td><td rowspan=1 colspan=1>82.9</td></tr><tr><td rowspan=1 colspan=1>Avg first 64 outputs</td><td rowspan=1 colspan=1>82.7</td></tr><tr><td rowspan=1 colspan=1>Avg all outputs</td><td rowspan=1 colspan=1>83.1</td></tr></table>",
+        "bbox": [
+            333,
+            397,
+            660,
+            465
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Bi-encoder results on the ConvAI2 valid set for different choices of function red(·). ",
+        "bbox": [
+            194,
+            476,
+            799,
+            491
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C Alternative Choices for Context Vectors ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            520,
+            545,
+            536
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We considered a few other ways to derive the context vectors $( y _ { c t x t } ^ { 1 } , . . . , y _ { c t x t } ^ { m } )$ of the Poly-encoder from the output $( h _ { c t x t } ^ { 1 } , . . . , h _ { c t x t } ^ { N } )$ of the underlying transformer: ",
+        "bbox": [
+            174,
+            550,
+            825,
+            580
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Learn $m$ codes $( c _ { 1 } , . . . , c _ { m } )$ , where $c _ { i }$ extracts representation $y _ { c t x t } ^ { i }$ by attending over all the outputs $( h _ { c t x t } ^ { 1 } , . . . , h _ { c t x t } ^ { N } )$ , . This method is denoted “Poly-encoder (Learnt-codes)” or “Poly, ...,encoder (Learnt-m)”, and is the method described in section 4.4   \n• Consider the first $m$ outputs $( h _ { c t x t } ^ { 1 } , . . . , h _ { c t x t } ^ { m } )$ . This method is denoted “Poly-encoder (First $m$ , ..., outputs)” or “Poly-encoder (First-m)”. Note that when $N \\ < \\ m$ , only $m$ vectors are considered.   \n• Consider the last $m$ outputs.   \n• Consider the last $m$ outputs concatenated with the first one, $h _ { c t x t } ^ { 1 }$ which plays a particular role in BERT as it corresponds to the special token [S]. ",
+        "bbox": [
+            215,
+            592,
+            825,
+            733
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The performance of those four methods is evaluated on the validation set of Convai2 and DSTC7 and reported on Table 8. The first two methods are shown in Figure 2. We additionally provide the inference time for a given number of candidates coming from the Convai2 dataset on Table 9. ",
+        "bbox": [
+            176,
+            744,
+            825,
+            786
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/a9b142c845a2eae0b130c3bb0635eecc4dfd6f0059c57cc92e78c77cfd9f4391.jpg",
+        "table_caption": [
+            "Table 8: Validation and test performance of Poly-encoder variants, with weights initialized from (Devlin et al., 2019). Scores are shown for ConvAI2 and DSTC 7 Track 1. Bold numbers indicate the highest performing variant within that number of codes. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=3>ConvAI2</td><td rowspan=1 colspan=1>DS</td><td rowspan=1 colspan=1>TC7</td></tr><tr><td rowspan=1 colspan=1>split</td><td rowspan=1 colspan=2>dev</td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1>test</td></tr><tr><td rowspan=1 colspan=1>metric</td><td rowspan=1 colspan=2>R@1/20</td><td rowspan=1 colspan=1>R@1/20</td><td rowspan=1 colspan=1>R@1/100</td><td rowspan=1 colspan=1>R@1/100</td></tr><tr><td rowspan=1 colspan=1>(Wolf et al., 2019)</td><td rowspan=1 colspan=2>82.1</td><td rowspan=1 colspan=1>80.7</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Chen &amp; Wang,2019)</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>57.3</td><td rowspan=1 colspan=1>64.5</td></tr><tr><td rowspan=1 colspan=3>1 Attention Code</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Learnt-codesFirst m outputsLast m outputsLast m outputs and hctxt</td><td rowspan=1 colspan=2>81.9 ± 0.383.2 ± 0.282.9 ± 0.1</td><td rowspan=1 colspan=1>81.0 ± 0.181.5 ± 0.181.0 ± 0.11</td><td rowspan=1 colspan=1>56.2 ± 0.156.4 ± 0.356.1 ± 0.41</td><td rowspan=1 colspan=1>66.9 ± 0.766.8 ± 0.767.2 ± 1.11</td></tr><tr><td rowspan=1 colspan=1>4 Attention Codes</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>Learnt-codesFirst m outputsLast m outputsLast m outputs and hctxt</td><td rowspan=2 colspan=2>83.8 ± 0.283.4 ± 0.282.8 ± 0.282.9 ± 0.1</td><td rowspan=2 colspan=1>82.2 ± 0.581.6 ± 0.181.3 ± 0.481.4 ± 0.2</td><td rowspan=1 colspan=1>56.5 ± 0.556.9 ± 0.556.0 ± 0.5</td><td rowspan=2 colspan=1>66.8 ± 0.767.2 ± 1.365.8 ± 0.566.1 ± 0.8</td></tr><tr><td rowspan=1 colspan=1>55.8 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>16 Attention Codes</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=3 colspan=1>Learnt-codesFirst m outputsLast m outputsLast m outputs and hctxt</td><td rowspan=3 colspan=2>84.4 ± 0.185.2 ± 0.183.9 ± 0.283.8 ± 0.3</td><td rowspan=2 colspan=1>83.2 ± 0.183.9 ± 0.282.0 ± 0.4</td><td rowspan=1 colspan=1>57.7 ± 0.2</td><td rowspan=1 colspan=1>67.8 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>56.1 ± 1.756.1 ± 0.3</td><td rowspan=1 colspan=1>66</td></tr><tr><td rowspan=1 colspan=1>81.7 ± 0.3</td><td rowspan=1 colspan=1>56.1 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>64 Attention Codes</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Learnt-codesFirst m outputsLast m outputsLast m outputs and hetxt</td><td rowspan=1 colspan=2>84.9 ± 0.186.0 ± 0.284.9 ± 0.385.0 ± 0.2</td><td rowspan=1 colspan=1>83.7 ± 0.284.2 ± 0.282.9 ± 0.283.2 ± 0.2</td><td rowspan=1 colspan=1>58.3 ± 0.457.7 ± 0.657.0 ± 0.257.3 ± 0.3</td><td rowspan=1 colspan=1>67.0± 0.967.1 ± 0.166.5 ± 0.567.1 ± 0.5</td></tr><tr><td rowspan=1 colspan=1>360 Attention Codes</td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=4 colspan=1>Learnt-codesFirst m outputsLast m outputsLast m outputs and hctxt</td><td rowspan=1 colspan=2>85.3 ± 0.3</td><td rowspan=1 colspan=1>83.7 ± 0.2</td><td rowspan=1 colspan=1>57.7 ± 0.3</td><td rowspan=1 colspan=1>68.9 ± 0.4</td></tr><tr><td rowspan=2 colspan=2>86.3 ± 0.186.3 ± 0.1</td><td rowspan=2 colspan=1>84.6 ± 0.384.7 ± 0.3</td><td rowspan=1 colspan=1>58.1 ± 0.4</td><td rowspan=2 colspan=1>66.8 ± 0.768.1 ± 0.5</td></tr><tr><td rowspan=2 colspan=2>86.3 ± 0.186.2 ± 0.3</td><td rowspan=1 colspan=1>86.3 ± 0.1</td><td rowspan=2 colspan=1>84.7 ± 0.384.5 ± 0.4</td><td rowspan=1 colspan=1>58.0 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>58.3 ± 0.4</td><td rowspan=1 colspan=1>68.0 ± 0.8</td></tr></table>",
+        "bbox": [
+            232,
+            132,
+            766,
+            569
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/3cd17fd25423767a34ab631e5365e56b906fc7b1b18d39bf37eaaf71baab50ac.jpg",
+        "table_caption": [
+            "Table 9: Average time in milliseconds to predict the next dialogue utterance from $N$ possible candidates. \\* are inferred. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=4>Scoring time (ms)</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>CPU</td><td rowspan=1 colspan=2>GPU</td></tr><tr><td rowspan=1 colspan=1>Candidates</td><td rowspan=1 colspan=1>1k</td><td rowspan=1 colspan=1>100k</td><td rowspan=1 colspan=1>1k</td><td rowspan=1 colspan=1>100k</td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>115</td><td rowspan=1 colspan=1>160</td><td rowspan=1 colspan=1>19</td><td rowspan=1 colspan=1>22</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First m outputs) 16</td><td rowspan=1 colspan=1>119</td><td rowspan=1 colspan=1>551</td><td rowspan=1 colspan=1>17</td><td rowspan=1 colspan=1>37</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First m outputs) 64</td><td rowspan=1 colspan=1>124</td><td rowspan=1 colspan=1>570</td><td rowspan=1 colspan=1>17</td><td rowspan=1 colspan=1>39</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First m outputs) 360</td><td rowspan=1 colspan=1>120</td><td rowspan=1 colspan=1>619</td><td rowspan=1 colspan=1>17</td><td rowspan=1 colspan=1>45</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-codes) 16</td><td rowspan=1 colspan=1>122</td><td rowspan=1 colspan=1>678</td><td rowspan=1 colspan=1>18</td><td rowspan=1 colspan=1>38</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-codes) 64</td><td rowspan=1 colspan=1>126</td><td rowspan=1 colspan=1>692</td><td rowspan=1 colspan=1>23</td><td rowspan=1 colspan=1>46</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-codes) 360</td><td rowspan=1 colspan=1>160</td><td rowspan=1 colspan=1>837</td><td rowspan=1 colspan=1>57</td><td rowspan=1 colspan=1>88</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>21.7k</td><td rowspan=1 colspan=1>2.2M*</td><td rowspan=1 colspan=1>2.6k</td><td rowspan=1 colspan=1>266k*</td></tr></table>",
+        "bbox": [
+            274,
+            698,
+            718,
+            848
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/9221b3a1d779860a34fcba41054b7511ced7276ff5b78615080559f193f9eaa8.jpg",
+        "image_caption": [
+            "Figure 2: (a) The Bi-encoder (b) The Cross-encoder (c) The Poly-encoder with first m vectors. (d) The Poly-encoder with $m$ learnt codes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            102,
+            794,
+            496
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/09b3e688158bc2e65d9e402bfa91cb5392e3cfecdde52cd3d9a81d7c441025e3.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=2>ConvAI2</td><td rowspan=1 colspan=4>DSTC7</td><td rowspan=1 colspan=4>Ubuntu v2</td></tr><tr><td rowspan=1 colspan=1>split</td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>test</td></tr><tr><td rowspan=1 colspan=1>metric</td><td rowspan=1 colspan=1>R@1/20</td><td rowspan=1 colspan=1>R@1/20</td><td rowspan=1 colspan=1>R@1/100</td><td rowspan=1 colspan=1>R@1/100</td><td rowspan=1 colspan=1>R@10/100</td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>R@1/10</td><td rowspan=1 colspan=1>R@1/10</td><td rowspan=1 colspan=1>R@5/10</td><td rowspan=1 colspan=1>MRR</td></tr><tr><td rowspan=1 colspan=1>HuggingFace(Wolf et al.,2019)</td><td rowspan=1 colspan=1>82.1</td><td rowspan=1 colspan=1>80.7</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>(Chen&amp;Wang,2019)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>57.3</td><td rowspan=1 colspan=1>64.5</td><td rowspan=1 colspan=1>90.2</td><td rowspan=1 colspan=1>73.5</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td></tr><tr><td rowspan=1 colspan=1>(Dong&amp;Huang,2018)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>75.9</td><td rowspan=1 colspan=1>97.3</td><td rowspan=1 colspan=1>84.8</td></tr><tr><td rowspan=1 colspan=1>pre-trained weights from(Dev</td><td rowspan=1 colspan=1>inetal.,201</td><td rowspan=1 colspan=1>9)-Toronto</td><td rowspan=1 colspan=1>Books+Wi</td><td rowspan=1 colspan=1>tipedia</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>83.3 ± 0.2</td><td rowspan=1 colspan=1>81.7 ± 0.2</td><td rowspan=1 colspan=1>56.5 ± 0.4</td><td rowspan=1 colspan=1>66.8 ± 0.7</td><td rowspan=1 colspan=1>89.0 ± 1.0</td><td rowspan=1 colspan=1>74.6 ± 0.5</td><td rowspan=1 colspan=1>80.9± 0.6</td><td rowspan=1 colspan=1>80.6 ± 0.4</td><td rowspan=1 colspan=1>98.2 ± 0.1</td><td rowspan=1 colspan=1>88.0 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 16</td><td rowspan=1 colspan=1>85.2 ± 0.1</td><td rowspan=1 colspan=1>83.9 ± 0.2</td><td rowspan=1 colspan=1>56.7 ± 0.2</td><td rowspan=1 colspan=1>67.0± 0.9</td><td rowspan=1 colspan=1>88.8±0.3</td><td rowspan=1 colspan=1>74.6± 0.6</td><td rowspan=1 colspan=1>81.7 ± 0.5</td><td rowspan=1 colspan=1>81.4 ± 0.6</td><td rowspan=1 colspan=1>98.2 ± 0.1</td><td rowspan=1 colspan=1>88.5 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder(Learnt-m) 16</td><td rowspan=1 colspan=1>84.4 ± 0.1</td><td rowspan=1 colspan=1>83.2 ± 0.1</td><td rowspan=1 colspan=1>57.7 ± 0.2</td><td rowspan=1 colspan=1>67.8± 0.3</td><td rowspan=1 colspan=1>88.6± 0.2</td><td rowspan=1 colspan=1>75.1 ± 0.2</td><td rowspan=1 colspan=1>81.5±0.1</td><td rowspan=1 colspan=1>81.2 ±0.2</td><td rowspan=1 colspan=1>98.2 ±0.0</td><td rowspan=1 colspan=1>88.3 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder(First-m) 64</td><td rowspan=1 colspan=1>86.0 ± 0.2</td><td rowspan=1 colspan=1>84.2 ± 0.2</td><td rowspan=1 colspan=1>57.1 ± 0.2</td><td rowspan=1 colspan=1>66.9 ± 0.7</td><td rowspan=1 colspan=1>89.1 ± 0.2</td><td rowspan=1 colspan=1>74.7 ± 0.4</td><td rowspan=1 colspan=1>82.2 ± 0.6</td><td rowspan=1 colspan=1>81.9 ± 0.5</td><td rowspan=1 colspan=1>98.4 ±0.0</td><td rowspan=1 colspan=1>88.8± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 64</td><td rowspan=1 colspan=1>84.9 ± 0.1</td><td rowspan=1 colspan=1>83.7 ± 0.2</td><td rowspan=1 colspan=1>58.3 ± 0.4</td><td rowspan=1 colspan=1>67.0± 0.9</td><td rowspan=1 colspan=1>89.2 ± 0.2</td><td rowspan=1 colspan=1>74.7 ± 0.6</td><td rowspan=1 colspan=1>81.8 ± 0.1</td><td rowspan=1 colspan=1>81.3 ± 0.2</td><td rowspan=1 colspan=1>98.2 ± 0.1</td><td rowspan=1 colspan=1>88.4 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 360</td><td rowspan=1 colspan=1>86.3 ± 0.1</td><td rowspan=1 colspan=1>84.6 ± 0.3</td><td rowspan=1 colspan=1>57.8 ± 0.5</td><td rowspan=1 colspan=1>67.0± 0.5</td><td rowspan=1 colspan=1>89.6± 0.9</td><td rowspan=1 colspan=1>75.0± 0.6</td><td rowspan=1 colspan=1>82.7 ± 0.4</td><td rowspan=1 colspan=1>82.2 ± 0.6</td><td rowspan=1 colspan=1>98.4±0.1</td><td rowspan=1 colspan=1>89.0 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 360</td><td rowspan=1 colspan=1>85.3 ± 0.3</td><td rowspan=1 colspan=1>83.7 ±0.2</td><td rowspan=1 colspan=1>57.7 ± 0.3</td><td rowspan=1 colspan=1>68.9 ± 0.4</td><td rowspan=1 colspan=1>89.9 ± 0.5</td><td rowspan=1 colspan=1>76.2 ± 0.2</td><td rowspan=1 colspan=1>81.5 ± 0.1</td><td rowspan=1 colspan=1>80.9 ± 0.1</td><td rowspan=1 colspan=1>98.1 ± 0.0</td><td rowspan=1 colspan=1>88.1 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>87.1 ± 0.1</td><td rowspan=1 colspan=1>84.8± 0.3</td><td rowspan=1 colspan=1>59.4 ± 0.4</td><td rowspan=1 colspan=1>67.4± 0.7</td><td rowspan=1 colspan=1>90.5 ± 0.3</td><td rowspan=1 colspan=1>75.6 ± 0.4</td><td rowspan=1 colspan=1>83.3 ± 0.4</td><td rowspan=1 colspan=1>82.8±0.3</td><td rowspan=1 colspan=1>98.4 ± 0.1</td><td rowspan=1 colspan=1>89.4 ± 0.2</td></tr><tr><td rowspan=1 colspan=1>Our pre-training on Toronto Books + Wikipedia</td><td rowspan=1 colspan=1>oks+Wiki</td><td rowspan=1 colspan=1>edia</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>84.6 ± 0.1</td><td rowspan=1 colspan=1>82.0 ± 0.1</td><td rowspan=1 colspan=1>54.9 ± 0.5</td><td rowspan=1 colspan=1>64.5 ± 0.5</td><td rowspan=1 colspan=1>88.1 ± 0.2</td><td rowspan=1 colspan=1>72.6 ± 0.4</td><td rowspan=1 colspan=1>80.9 ± 0.5</td><td rowspan=1 colspan=1>80.8 ± 0.5</td><td rowspan=1 colspan=1>98.4 ± 0.1</td><td rowspan=1 colspan=1>88.2 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 16</td><td rowspan=1 colspan=1>84.1 ± 0.2</td><td rowspan=1 colspan=1>81.4 ± 0.2</td><td rowspan=1 colspan=1>53.9 ± 2.7</td><td rowspan=1 colspan=1>63.3 ± 2.9</td><td rowspan=1 colspan=1>87.2 ± 1.5</td><td rowspan=1 colspan=1>71.6 ± 2.4</td><td rowspan=1 colspan=1>80.8 ± 0.5</td><td rowspan=1 colspan=1>80.6 ±0.4</td><td rowspan=1 colspan=1>98.4 ± 0.1</td><td rowspan=1 colspan=1>88.1 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 16</td><td rowspan=1 colspan=1>85.4± 0.2</td><td rowspan=1 colspan=1>82.7 ±0.1</td><td rowspan=1 colspan=1>56.0± 0.4</td><td rowspan=1 colspan=1>65.3± 0.9</td><td rowspan=1 colspan=1>88.2± 0.7</td><td rowspan=1 colspan=1>73.2 ± 0.7</td><td rowspan=1 colspan=1>84.0± 0.1</td><td rowspan=1 colspan=1>83.4 ±0.2</td><td rowspan=1 colspan=1>98.7±0.0</td><td rowspan=1 colspan=1>89.9 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder(First-m) 64</td><td rowspan=1 colspan=1>86.1 ±0.4</td><td rowspan=1 colspan=1>83.9± 0.3</td><td rowspan=1 colspan=1>55.6 ± 0.9</td><td rowspan=1 colspan=1>64.3 ± 1.5</td><td rowspan=1 colspan=1>87.8 ± 0.4</td><td rowspan=1 colspan=1>72.5 ± 1.0</td><td rowspan=1 colspan=1>80.9 ± 0.6</td><td rowspan=1 colspan=1>80.7 ± 0.6</td><td rowspan=1 colspan=1>98.4± 0.0</td><td rowspan=1 colspan=1>88.2 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 64</td><td rowspan=1 colspan=1>85.6 ± 0.1</td><td rowspan=1 colspan=1>83.3 ± 0.1</td><td rowspan=1 colspan=1>56.2 ± 0.4</td><td rowspan=1 colspan=1>65.8 ± 0.7</td><td rowspan=1 colspan=1>88.4 ± 0.3</td><td rowspan=1 colspan=1>73.5 ± 0.5</td><td rowspan=1 colspan=1>84.0 ± 0.1</td><td rowspan=1 colspan=1>83.4 ± 0.1</td><td rowspan=1 colspan=1>98.7 ±0.0</td><td rowspan=1 colspan=1>89.9 ± 0.0</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 360</td><td rowspan=1 colspan=1>86.6± 0.3</td><td rowspan=1 colspan=1>84.4 ± 0.2</td><td rowspan=1 colspan=1>57.5 ± 0.4</td><td rowspan=1 colspan=1>66.5 ± 1.2</td><td rowspan=1 colspan=1>89.0 ± 0.5</td><td rowspan=1 colspan=1>74.4 ± 0.7</td><td rowspan=1 colspan=1>81.3±0.6</td><td rowspan=1 colspan=1>81.1 ±0.4</td><td rowspan=1 colspan=1>98.4 ± 0.2</td><td rowspan=1 colspan=1>88.4 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder(Learnt-m) 360</td><td rowspan=1 colspan=1>86.1 ± 0.1</td><td rowspan=1 colspan=1>83.8 ± 0.1</td><td rowspan=1 colspan=1>56.5 ± 0.8</td><td rowspan=1 colspan=1>65.8 ± 0.7</td><td rowspan=1 colspan=1>88.5 ± 0.6</td><td rowspan=1 colspan=1>73.6 ± 0.6</td><td rowspan=1 colspan=1>84.2 ± 0.2</td><td rowspan=1 colspan=1>83.7 ±0.0</td><td rowspan=1 colspan=1>98.7 ± 0.1</td><td rowspan=1 colspan=1>90.1 ± 0.0</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>87.3 ± 0.5</td><td rowspan=1 colspan=1>84.9 ± 0.3</td><td rowspan=1 colspan=1>57.7 ± 0.5</td><td rowspan=1 colspan=1>65.3 ± 1.0</td><td rowspan=1 colspan=1>89.7 ± 0.5</td><td rowspan=1 colspan=1>73.8 ± 0.6</td><td rowspan=1 colspan=1>83.2 ±0.8</td><td rowspan=1 colspan=1>83.1 ± 0.7</td><td rowspan=1 colspan=1>98.7 ± 0.1</td><td rowspan=1 colspan=1>89.7 ± 0.5</td></tr><tr><td rowspan=1 colspan=1>Our pre-training on Reddit</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Bi-encoder</td><td rowspan=1 colspan=1>86.9 ± 0.1</td><td rowspan=1 colspan=1>84.8 ± 0.1</td><td rowspan=1 colspan=1>60.1 ± 0.4</td><td rowspan=1 colspan=1>70.9 ± 0.5</td><td rowspan=1 colspan=1>90.6 ± 0.3</td><td rowspan=1 colspan=1>78.1 ± 0.3</td><td rowspan=1 colspan=1>83.7±0.7</td><td rowspan=1 colspan=1>83.6 ±0.7</td><td rowspan=1 colspan=1>98.8±0.1</td><td rowspan=1 colspan=1>90.1 ± 0.4</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder(First-m) 16</td><td rowspan=1 colspan=1>89.0± 0.1</td><td rowspan=1 colspan=1>86.4± 0.3</td><td rowspan=1 colspan=1>60.4± 0.3</td><td rowspan=1 colspan=1>70.7 ± 0.7</td><td rowspan=1 colspan=1>91.0± 0.4</td><td rowspan=1 colspan=1>78.0± 0.5</td><td rowspan=1 colspan=1>84.3 ± 0.3</td><td rowspan=1 colspan=1>84.3± 0.2</td><td rowspan=1 colspan=1>98.9± 0.0</td><td rowspan=1 colspan=1>90.5 ±0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 16</td><td rowspan=1 colspan=1>88.6 ± 0.3</td><td rowspan=1 colspan=1>86.3 ± 0.3</td><td rowspan=1 colspan=1>61.1 ± 0.4</td><td rowspan=1 colspan=1>71.6 ± 0.6</td><td rowspan=1 colspan=1>91.3 ± 0.3</td><td rowspan=1 colspan=1>78.4 ± 0.4</td><td rowspan=1 colspan=1>86.1 ± 0.1</td><td rowspan=1 colspan=1>86.0 ± 0.1</td><td rowspan=1 colspan=1>99.0 ± 0.1</td><td rowspan=1 colspan=1>91.5 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 64</td><td rowspan=1 colspan=1>89.5 ± 0.1</td><td rowspan=1 colspan=1>87.3 ± 0.2</td><td rowspan=1 colspan=1>61.0 ± 0.4</td><td rowspan=1 colspan=1>70.9 ± 0.6</td><td rowspan=1 colspan=1>91.5 ± 0.5</td><td rowspan=1 colspan=1>78.0± 0.3</td><td rowspan=1 colspan=1>84.0 ± 0.4</td><td rowspan=1 colspan=1>83.9 ± 0.4</td><td rowspan=1 colspan=1>98.8±0.0</td><td rowspan=1 colspan=1>90.3 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 64</td><td rowspan=1 colspan=1>89.0± 0.1</td><td rowspan=1 colspan=1>86.5 ± 0.2</td><td rowspan=1 colspan=1>60.9± 0.6</td><td rowspan=1 colspan=1>71.2 ± 0.8</td><td rowspan=1 colspan=1>91.3± 0.4</td><td rowspan=1 colspan=1>78.2± 0.7</td><td rowspan=1 colspan=1>86.2 ± 0.1</td><td rowspan=1 colspan=1>85.9 ± 0.1</td><td rowspan=1 colspan=1>99.1 ± 0.0</td><td rowspan=1 colspan=1>91.5 ± 0.1</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (First-m) 360</td><td rowspan=1 colspan=1>90.0 ± 0.1</td><td rowspan=1 colspan=1>87.3 ± 0.1</td><td rowspan=1 colspan=1>61.1 ± 1.9</td><td rowspan=1 colspan=1>70.9 ± 2.1</td><td rowspan=1 colspan=1>91.5 ± 0.9</td><td rowspan=1 colspan=1>77.9 ± 1.6</td><td rowspan=1 colspan=1>84.8 ± 0.5</td><td rowspan=1 colspan=1>84.6 ± 0.5</td><td rowspan=1 colspan=1>98.9 ± 0.1</td><td rowspan=1 colspan=1>90.7 ± 0.3</td></tr><tr><td rowspan=1 colspan=1>Poly-encoder (Learnt-m) 360</td><td rowspan=1 colspan=1>89.2 ± 0.1</td><td rowspan=1 colspan=1>86.8± 0.1</td><td rowspan=1 colspan=1>61.2 ± 0.2</td><td rowspan=1 colspan=1>71.4 ± 1.0</td><td rowspan=1 colspan=1>91.1 ± 0.3</td><td rowspan=1 colspan=1>78.3± 0.7</td><td rowspan=1 colspan=1>86.3 ± 0.1</td><td rowspan=1 colspan=1>85.9 ± 0.1</td><td rowspan=1 colspan=1>99.1 ±0.0</td><td rowspan=1 colspan=1>91.5 ± 0.0</td></tr><tr><td rowspan=1 colspan=1>Cross-encoder</td><td rowspan=1 colspan=1>90.3± 0.2</td><td rowspan=1 colspan=1>87.9 ± 0.2</td><td rowspan=1 colspan=1>63.9 ± 0.3</td><td rowspan=1 colspan=1>71.7 ± 0.3</td><td rowspan=1 colspan=1>92.4 ± 0.5</td><td rowspan=1 colspan=1>79.0 ± 0.2</td><td rowspan=1 colspan=1>86.7±0.1</td><td rowspan=1 colspan=1>86.5 ± 0.1</td><td rowspan=1 colspan=1>99.1 ± 0.0</td><td rowspan=1 colspan=1>91.9 ± 0.0</td></tr></table>",
+        "bbox": [
+            171,
+            550,
+            825,
+            878
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 10: Validation and test performances of Bi-, Poly- and Cross-encoders. Scores are shown for ConvAI2, DSTC7 Track 1 and Ubuntu v2, and the previous state-of-the-art models in the literature. ",
+        "bbox": [
+            173,
+            888,
+            825,
+            917
+        ],
+        "page_idx": 13
+    }
+]

parse/train/TiwPYwg3IRf/TiwPYwg3IRf.md

@@ -0,0 +1,325 @@
+# Generalized Jensen-Shannon Divergence Loss for Learning with Noisy Labels
+
+Erik Englesson KTH Stockholm, Sweden engless@kth.se
+
+Hossein Azizpour KTH Stockholm, Sweden azizpour@kth.se
+
+# Abstract
+
+Prior works have found it beneficial to combine provably noise-robust loss functions e.g., mean absolute error (MAE) with standard categorical loss function e.g. cross entropy (CE) to improve their learnability. Here, we propose to use Jensen-Shannon divergence as a noise-robust loss function and show that it interestingly interpolate between CE and MAE with a controllable mixing parameter. Furthermore, we make a crucial observation that CE exhibits lower consistency around noisy data points. Based on this observation, we adopt a generalized version of the JensenShannon divergence for multiple distributions to encourage consistency around data points. Using this loss function, we show state-of-the-art results on both synthetic (CIFAR), and real-world (e.g. WebVision) noise with varying noise rates.
+
+# 1 Introduction
+
+Labeled datasets, even the systematically annotated ones, contain noisy labels [1]. Therefore, designing noise-robust learning algorithms are crucial for the real-world tasks. An important avenue to tackle noisy labels is to devise noise-robust loss functions [2, 3, 4, 5]. Similarly, in this work, we propose two new noise-robust loss functions based on two central observations as follows.
+
+Observation I: Provably-robust loss functions can underfit the training data [2, 3, 4, 5].   
+Observation II: Standard networks show low consistency around noisy data points 1, see Figure 1.
+
+We first propose to use Jensen-Shannon divergence (JS) as a loss function, which we crucially show interpolates between the noise-robust mean absolute error (MAE) and the cross entropy (CE) that better fits the data through faster convergence. Figure 2 illustrates the CE-MAE interpolation. Regarding Observation II, we adopt the generalized version of Jensen-Shannon divergence (GJS) to encourage predictions on perturbed inputs to be consistent, see Figure 3. Notably, Jensen-Shannon divergence has previously shown promise for test-time robustness to domain shift [6], here we further argue for its training-time robustness to label noise. The key contributions of this work2 are:
+
+• We make a novel observation that a network predictions’ consistency is reduced for noisylabeled data when overfitting to noise, which motivates the use of consistency regularization. • We propose using Jensen-Shannon divergence (JS) and its multi-distribution generalization (GJS) as loss functions for learning with noisy labels. We relate JS to loss functions that are based on the noise-robustness theory of Ghosh et al. [2]. In particular, we prove that JS generalizes CE and MAE. Furthermore, we prove that GJS generalizes JS by incorporating consistency regularization in a single principled loss function. • We provide an extensive set of empirical evidences on several datasets, noise types and rates. They show state-of-the-art results and give in-depth studies of the proposed losses.
+
+![](images/7c9ce0e5ce3b6ce01768b4299d109e57f0d618594199d5247062bc1e2f73ea34.jpg)  
+Figure 1: Evolution of a trained network’s consistency as it overfits to noise using CE loss. Here we plot the evolution of the validation accuracy (a) and network’s consistency (as measured by GJS) on clean (b) and noisy (c) examples of the training set of CIFAR-100 for varying symmetric noise rates when learning with the cross-entropy loss. The consistency of the learnt function and the accuracy closely correlate. This suggests that enforcing consistency may help avoid fitting to noise. Furthermore, the consistency is degraded more significantly for the noisy data points.
+
+# 2 Generalized Jensen-Shannon Divergence
+
+We propose two loss functions, the Jensen-Shannon divergence (JS) and its multi-distribution generalization (GJS). In this section, we first provide background and two observations that motivate our proposed loss functions. This is followed by definition of the losses, and then we show that JS generalizes CE and MAE similarly to other robust loss functions. Finally, we show how GJS generalizes JS to incorporate consistency regularization into a single principled loss function. We provide proofs of all theorems, propositions, and remarks in this section in Appendix C.
+
+# 2.1 Background & Motivation
+
+Supervised Classification. Assume a general function class3 $\mathcal { F }$ where each $f \in \mathcal F$ maps an input $\textbf { \em x } \in \mathrm { ~ \mathbb ~ X ~ }$ to the probability simplex $\Delta ^ { K - 1 }$ , i.e. to a categorical distribution over $K$ classes $\boldsymbol { y } \in \mathbb { Y } = \{ 1 , 2 , \dots , K \}$ . We seek $f ^ { * } \in { \mathcal { F } }$ that minimizes a risk $R _ { \mathcal { L } } ( f ) = \mathbb { E } _ { \mathcal { D } } [ \mathcal { L } ( e ^ { ( y ) } , f ( \pmb { x } ) ) ]$ , for some loss function $\mathcal { L }$ and joint distribution $\mathcal { D }$ over $\mathbb { X } \times \mathbb { Y }$ , where $e ^ { ( y ) }$ is a $K$ -vector with one at index and zero elsewhere. In practice, $\mathcal { D }$ is unknown and, instead, we use ${ \cal S } = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ which are independently sampled from $\mathcal { D }$ to minimize an empirical risk $\begin{array} { r } { \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathcal { L } ( e ^ { ( y _ { i } ) } , f ( \pmb { x } _ { i } ) ) } \end{array}$ =1.
+
+Learning with Noisy Labels. In this work, the goal is to learn from a noisy training distribution $\mathcal { D } _ { \eta }$ where the labels are changed, with probability $\eta$ , from their true distribution $\mathcal { D }$ . The noise is called instance-dependent if it depends on the input, asymmetric if it dependents on the true label, and symmetric if it is independent of both $_ { \textbf { \em x } }$ and $y$ . Let $f _ { \eta } ^ { * }$ be the optimizer of the noisy distribution risk $R _ { \mathcal { L } } ^ { \eta } ( f )$ . A loss function $\mathcal { L }$ is then called robust if $f _ { \eta } ^ { * }$ also minimizes $R _ { \mathcal { L } }$ . The MAE loss $( \mathcal L _ { M A E } ( e ^ { ( y ) } , f ( \pmb x ) ) : = \| e ^ { ( y ) } - f ( \pmb x ) \| _ { 1 } )$ is robust but not CE [2].
+
+Issue of Underfitting. Several works propose such robust loss functions and demonstrate their efficacy in preventing noise fitting [2, 3, 4, 5]. However, all those works have observed slow convergence of such robust loss functions leading to underfitting. This can be contrasted with CE that has fast convergence but overfits to noise. Ghosh et al. [2] mentions slow convergence of MAE and GCE [3] extensively analyzes the undefitting thereof. SCE [4] reports similar problems for the reverse cross entropy and proposes a linear combination with CE. Finally, Ma et al. [5] observe the same problem and consider a combination of “active” and “passive” loss functions.
+
+Consistency Regularization. This encourages a network to have consistent predictions for different perturbations of the same image, which has mainly been used for semi-supervised learning [7].
+
+Motivation. In Figure 1, we show the validation accuracy and a measure of consistency during training with the CE loss for varying amounts of noise. First, we note that training with CE loss eventually overfits to noisy labels. Figure 1a, indicates that the higher the noise rate, the more accuracy drop when it starts to overfit to noise. Figure 1(b-c) shows the consistency of predictions for correct and noisy labeled examples of the training set, with the consistency measured as the ratio of examples that have the same class prediction for two perturbations of the same image, see
+
+![](images/06fcb0a20948d227fbd1a3afea7ab398020ba3f9dd73b458205a3ff9e6a6822d.jpg)  
+Figure 2: JS loss generalizes CE and MAE. The Jensen-Shannon loss $( \mathcal { L } _ { \mathrm { J S } } )$ for different values of the hyperparameter $\pi _ { 1 }$ . The JS loss interpolates between CE and MAE. For low values of $\pi _ { 1 }$ , ${ \mathcal { L } } _ { \mathrm { J S } }$ behaves like CE and for increasing values of $\pi _ { 1 }$ it behaves more like the noise robust MAE loss.
+
+![](images/5e65c0bbb5a37ec95fdbf5a65f0b12ec9dae92e8dd123a385e8f603049989121.jpg)  
+Figure 3: GJS Dissection for $\mathbf { M } = \mathbf { K } = 3 \colon$ The decomposition of ${ \mathcal { L } } _ { \mathrm { G J S } }$ (left) into a JS term (middle) and a consistency term (right) from Proposition 2. Each point in the simplex correspond to a $\pmb { p } ^ { ( 3 ) } \in \Delta ^ { 2 }$ , where the color represents the value of the loss at that point. It can be seen that there are two ways to minimize ${ \mathcal { L } } _ { \mathrm { G J S } }$ , either by making the predictions similar to the label (middle) or similar to the other predictions (right) to increase consistency. To better highlight the variations of the losses, each loss has its own range of values.
+
+Appendix B.6 for more details. A clear correlation is observed between the accuracy and consistency of the noisy examples. This suggests that maximizing consistency of predictions may improve the robustness to noise. Next, we define simple loss functions that (i) encourage consistency around data points and (ii) alleviate the “issue of underfitting” by interpolating between CE and MAE.
+
+# 2.2 Definitions
+
+$D _ { \mathbf { J } \mathbf { S } }$ . Let $\pmb { p } ^ { ( 1 ) } , \pmb { p } ^ { ( 2 ) } \in \Delta ^ { K - 1 }$ have corresponding weights $\pmb { \pi } = [ \pi _ { 1 } , \pi _ { 2 } ] ^ { T } \in \Delta$ . Then, the JensenShannon divergence between $ { \mathbf { \mathit { p } } } ^ { ( 1 ) }$ and $ { p ^ { ( 2 ) } }$ is
+
+$D _ { 1 \mathrm { S } _ { \pi } } ( p ^ { ( 1 ) } , p ^ { ( 2 ) } ) : = H ( m ) - \pi _ { 1 } H ( p ^ { ( 1 ) } ) - \pi _ { 2 } H ( p ^ { ( 2 ) } ) = \pi _ { 1 } D _ { \mathrm { K L } } ( p ^ { ( 1 ) } | | m ) + \pi _ { 2 } D _ { \mathrm { K L } } ( p ^ { ( 2 ) } | | m )$ (1) with $H$ the Shannon entropy, and $\pmb { m } ~ = ~ \pi _ { 1 } \pmb { p } ^ { ( 1 ) } + \pi _ { 2 } \pmb { p } ^ { ( 2 ) }$ . Unlike Kullback–Leibler divergence $( D _ { \mathrm { K L } } ( \pmb { p } ^ { ( 1 ) } \| \pmb { p } ^ { ( 2 ) } ) )$ or cross entropy (CE), JS is symmetric, bounded, does not require absolute continuity, and has a crucial weighting mechanism $( \pi )$ , as we will see later.
+
+$D _ { \mathbf { G J S } }$ . Similar to $D _ { \mathrm { K L } }$ , $D _ { \mathrm { J S } }$ satisfies $D _ { \mathrm { J S } _ { \pi } } ( { p } ^ { ( 1 ) } , { p } ^ { ( 2 ) } ) \geq 0$ , with equality iff $\pmb { p } ^ { ( 1 ) } = \pmb { p } ^ { ( 2 ) }$ . For $D _ { \mathrm { J S } }$ this is derived from Jensen’s inequality for the concave Shannon entropy. This property holds for finite number of distributions and motivates a generalization of $D _ { \mathrm { J S } }$ to multiple distributions [8]:
+
+$$
+D _ { \mathrm { G J S } _ { \pi } } ( p ^ { ( 1 ) } , \dots , p ^ { ( M ) } ) : = H { \Bigl ( } \sum _ { i = 1 } ^ { M } \pi _ { i } p ^ { ( i ) } { \Bigr ) } - \sum _ { i = 1 } ^ { M } \pi _ { i } H ( p ^ { ( i ) } ) = \sum _ { i = 1 } ^ { M } \pi _ { i } D _ { \mathrm { K L } } { \Bigl ( } p ^ { ( i ) } { \Bigr \| } \sum _ { j = 1 } ^ { M } \pi _ { j } p ^ { ( j ) } { \Bigr ) }
+$$
+
+where $M$ is the number of distributions, and $\pmb { \pi } = [ \pi _ { 1 } , \ldots , \pi _ { M } ] ^ { T } \in \Delta ^ { M - 1 }$
+
+Loss functions. We aim to use $D _ { \mathrm { J S } }$ and $D _ { \mathrm { G J S } }$ divergences, to measure deviation of the predictive distribution(s), $f ( { \pmb x } )$ , from the target distribution, $e ^ { ( y ) }$ . Without loss of generality, hereafter, we dedicate $ { \mathbf { \mathit { p } } } ^ { ( 1 ) }$ to denote the target distribution. JS loss, therefore, can take the form of $D _ { \mathrm { J S } _ { \pi } } ( e ^ { ( y ) } , f ( { \pmb x } ) )$ . Generalized JS loss is a less straight-forward construction since $D _ { \mathrm { G J S } }$ can accommodate more predictive distributions. While various choices can be made for these distributions, in this work, we consider predictions associated with different random perturbations of a sample, denoted by $\scriptstyle A ( { \pmb x } )$ . This choice, as shown later, implies an interesting analogy to consistency regularization. The choice, also entails no distinction between the $M - 1$ predictive distributions. Therefore, we consider $\pi _ { 2 } = \cdot \cdot \cdot = \pi _ { M } = { \textstyle { \frac { 1 - \pi _ { 1 } } { M - 1 } } }$ in all our experiments. Finally, we scale the loss functions by a constant factor $Z = - ( 1 - \pi _ { 1 } ) \log ( 1 - \pi _ { 1 } )$ . As we will see later, the role of this scaling is merely to strengthen the already existing and desirable behaviors of these losses as $\pi _ { 1 }$ approaches zero and one. Formally, we have JS and GJS losses:
+
+$$
+\mathcal { L } _ { \mathrm { J S } } ( y , f , x ) : = \frac { D _ { \mathrm { J S } _ { \pi } } ( e ^ { ( y ) } , f ( \tilde { x } ) ) } { Z } , \quad \mathcal { L } _ { \mathrm { G J S } } ( y , f , x ) : = \frac { D _ { \mathrm { G J S } _ { \pi } } ( e ^ { ( y ) } , f ( \tilde { x } ^ { ( 2 ) } ) , \dots , f ( \tilde { x } ^ { ( M ) } ) ) } { Z }
+$$
+
+with $\tilde { \mathbf { x } } ^ { ( i ) } \sim \mathcal { A } ( \mathbf { x } )$ . Next, we study the connection between JS and losses which are based on the robustness theory of Ghosh et al. [2].
+
+# 2.3 JS’s Connection to Robust Losses
+
+Cross Entropy (CE) is the prevalent loss function for deep classifiers with remarkable successes. However, CE is prone to fitting noise [9]. On the other hand, Mean Absolute Error (MAE) is theoretically noise-robust [2]. Evidently, standard optimization algorithms struggle to minimize MAE, especially for more challenging datasets e.g. CIFAR-100 [3, 5]. Therefore, there have been several proposals that combine CE and MAE, such as Generalized CE (GCE) [3], Symmetric CE (SCE) [4], and Normalized CE (NCE+MAE) [5]. The rationale is for CE to help with the learning dynamics of MAE. Next, we show JS has CE and MAE as its asymptotes w.r.t. $\pi _ { 1 }$ .
+
+Proposition 1. Let $\pmb { p } \in \Delta ^ { K - 1 }$ , then
+
+$$
+\operatorname* { l i m } _ { \pi _ { 1 } \to 0 } \mathcal L _ { \mathrm { J S } } ( e ^ { ( y ) } , p ) = H ( e ^ { ( y ) } , p ) , \qquad \operatorname* { l i m } _ { \pi _ { 1 } \to 1 } \mathcal L _ { \mathrm { J S } } ( e ^ { ( y ) } , p ) = \frac { 1 } { 2 } \| e ^ { ( y ) } - p \| _ { 1 }
+$$
+
+where $H ( e ^ { ( y ) } , p )$ is the cross entropy of $e ^ { ( y ) }$ relative to $\pmb { p }$
+
+Figure 2 depicts how JS interpolates between CE and MAE for $\pi _ { 1 } \in ( 0 , 1 )$ . The proposition reveals an interesting connection to state-of-the-art robust loss functions, however, there are important differences. SCE is not bounded (so it cannot be used in Theorem 1), and GCE is not symmetric, while JS and MAE are both symmetric and bounded. In Appendix B.3, we perform a dissection to better understand how these properties affect learning with noisy labels. GCE is most similar to JS and is compared further in Appendix B.4.
+
+A crucial difference to these other losses is that JS naturally extends to multiple predictive distributions (GJS). Next, we show how GJS generalizes JS by incorporating consistency regularization.
+
+# 2.4 GJS’s Connection to Consistency Regularization
+
+In Figure 1, it was shown how the consistency of the noisy labeled examples was reduced when the network overfitted to noise. The following proposition shows how GJS naturally encourages consistency in a single principled loss function.
+
+Proposition 2. Let $\pmb { p } ^ { ( 2 ) } , \ldots , \pmb { p } ^ { ( M ) } \in \Delta ^ { K - 1 }$ with $M \geq 3$ and $\begin{array} { r } { \bar { p } _ { > 1 } = \frac { \sum _ { j = 2 } ^ { M } \pi _ { j } { p ^ { ( j ) } } } { 1 - \pi _ { 1 } } } \end{array}$ , then
+
+$$
+\mathscr { L } _ { \mathrm { G J S } } ( e ^ { ( y ) } , \pmb { p } ^ { ( 2 ) } , \dots , \pmb { p } ^ { ( M ) } ) = \mathscr { L } _ { \mathrm { J S } _ { \pi ^ { \prime } } } ( e ^ { ( y ) } , \bar { \pmb { p } } _ { > 1 } ) + ( 1 - \pi _ { 1 } ) \mathscr { L } _ { \mathrm { G J S } _ { \pi ^ { \prime \prime } } } ( \pmb { p } ^ { ( 2 ) } , \dots , \pmb { p } ^ { ( M ) } )
+$$
+
+where π0 = [π1, 1 − π1]T and π00 = [π2,...,πM ]T .
+
+Importantly, Proposition 2 shows that GJS can be decomposed into two terms: 1) a JS term between the label and the mean prediction $\bar { p } _ { > 1 }$ , and 2) a GJS term, but without the label. Figure 3 illustrates the effect of this decomposition. The first term, similarly to the standard JS loss, encourages the predictions’ mean to be closer to the label (Figure 3 middle). However, the second term encourages all predictions to be similar, that is, consistency regularization (Figure 3 right).
+
+# 2.5 Noise Robustness
+
+Here, the robustness properties of JS and GJS are analyzed in terms of lower $( B _ { L } )$ and upper bounds $( B _ { U } )$ for the following theorem, which generalizes the results by Zhang et al. [3] to any bounded loss function, even with multiple predictive distributions.
+
+Theorem 1. Undsatisfied for a loss symmetric noise with , then $\begin{array} { r } { \eta < \frac { K - 1 } { K } } \end{array}$ , $\begin{array} { r } { \{ B _ { L } \leq \sum _ { i = 1 } ^ { K } \mathcal { L } ( e ^ { ( i ) } , { \pmb x } , f ) \leq B _ { U } , \forall { \pmb x } , f } \end{array}$ $\mathcal { L }$
+
+$$
+0 \leq R _ { \mathcal { L } } ^ { \eta } ( f ^ { * } ) - R _ { \mathcal { L } } ^ { \eta } ( f _ { \eta } ^ { * } ) \leq \eta \frac { B _ { U } - B _ { L } } { K - 1 } , \quad a n d \quad - \frac { \eta ( B _ { U } - B _ { L } ) } { K - 1 - \eta K } \leq R _ { \mathcal { L } } ( f ^ { * } ) - R _ { \mathcal { L } } ( f _ { \eta } ^ { * } ) \leq 0 ,
+$$
+
+A tighter bound $B _ { U } - B _ { L }$ , implies a smaller worst case risk difference of the optimal classifiers (robust when $B _ { U } = B _ { L }$ ). Importantly, while $\mathcal { L } ( e ^ { ( i ) } , \pmb { x } , f ) = \mathcal { L } ( e ^ { ( i ) } , f ( \pmb { x } ) )$ usually, this subtle distinction is useful for losses with multiple predictive distributions, see Equation 3. In Theorem 2 in Appendix C.3, we further prove the robustness of the proposed losses to asymmetric noise.
+
+For losses with multiple predictive distributions, the bounds in Theorem 1 and 2 must hold for any $_ { \textbf { \em x } }$ and $f$ , i.e., for any combination of $M - 1$ categorical distributions on $K$ classes. Proposition 3 provides such bounds for GJS.
+
+Proposition 3. GJS loss with $M \leq K + 1$ satisfies $\begin{array} { r } { B _ { L } \le \sum _ { k = 1 } ^ { K } \mathcal { L } _ { \mathrm { G J S } } ( e ^ { ( k ) } , \pmb { p } ^ { ( 2 ) } , \dots , \pmb { p } ^ { ( M ) } ) \le B _ { U } } \end{array}$ for all $\pmb { p } ^ { ( 2 ) } , \ldots , \pmb { p } ^ { ( M ) } \in \Delta ^ { K - 1 }$ , with the following bounds
+
+$$
+B _ { L } = \sum _ { k = 1 } ^ { K } \mathcal { L } _ { \mathrm { G J S } } ( e ^ { ( k ) } , { \pmb u } , \ldots , { \pmb u } ) , \quad B _ { U } = \sum _ { k = 1 } ^ { K } \mathcal { L } _ { \mathrm { G J S } } ( e ^ { ( k ) } , e ^ { ( 1 ) } , \ldots , e ^ { ( M - 1 ) } )
+$$
+
+where $\pmb { u } \in \Delta ^ { K - 1 }$ is the uniform distribution.
+
+Note the bounds for the JS loss is a special case of Proposition 3 for $M = 2$
+
+Remark 1. ${ \mathcal { L } } _ { \mathrm { J S } }$ and ${ \mathcal { L } } _ { \mathrm { G J S } }$ are robust $B _ { L } = B _ { U , }$ ) in the limit of $\pi _ { 1 }  1$
+
+Remark 1 is intuitive from Section 2.3 which showed that ${ \mathcal { L } } _ { \mathrm { J S } }$ is equivalent to the robust MAE in this limit and that the consistency term in Proposition 2 vanishes.
+
+In Proposition 3, the lower bound $( B _ { L } )$ is the same for JS and GJS. However, the upper bound $( B _ { U } )$ increases for more distributions, which makes JS have a tighter bound than GJS in Theorem 1 and 2. In Proposition 4, we show that JS and GJS have the same bound for the risk difference, given an assumption based on Figure 1 that the optimal classifier on clean data $( f ^ { * } )$ is at least as consistent as the optimal classifier on noisy data $( f _ { \eta } ^ { * } )$ .
+
+${ \mathcal { L } } _ { \mathrm { J S } }$ ${ \mathcal { L } } _ { \mathrm { G J S } }$ ounds , where $^ { l }$ i f is $\mathbb { E } _ { \mathbf { x } } [ \mathcal { L } _ { \mathrm { G J S } _ { \pi ^ { \prime \prime } } } ^ { f ^ { * } } ( \pmb { p } ^ { ( 2 ) } , \dots , \pmb { p } ^ { ( M ) } ) ] \le \mathbb { E } _ { \mathbf { x } } [ \mathcal { L } _ { \mathrm { G J S } _ { \pi ^ { \prime \prime } } } ^ { f _ { \eta } ^ { * } } ( \pmb { p } ^ { ( 2 ) } , \dots , \pmb { p } ^ { ( M ) } ) ]$ $\mathcal { L } _ { \mathrm { G J S } _ { \pi ^ { \prime \prime } } } ^ { f } ( \pmb { p } ^ { ( 2 ) } , \dots , \pmb { p } ^ { ( M ) } )$ the consistency term from Proposition 2.
+
+# 3 Related Works
+
+Interleaved in the previous sections, we covered most-related works to us, i.e. the avenue of identification or construction of theoretically-motivated robust loss functions [2, 3, 4, 5]. These works, similar to this paper, follow the theoretical construction of Ghosh et al. [2]. Furthermore, Liu&Guo [10] use “peer prediction” to propose a new family of robust loss functions. Different to these works, here, we propose loss functions based on $D _ { \mathrm { J S } }$ which holds various desirable properties of those prior works while exhibiting novel ties to consistency regularization; a recent important regularization technique.
+
+Next, we briefly cover other lines of work. A more thorough version can be found in Appendix D.
+
+A direction, that similar to us does not alter training, reweights a loss function by confusion matrix [11, 12, 13, 14, 15]. Assuming a class-conditional noise model, loss correction is theoretically motivated and perfectly orthogonal to noise-robust losses.
+
+Consistency regularization is a recent technique that imposes smoothness in the learnt function for semi-supervised learning [7] and recently for noisy data [16]. These works use different complex pipelines for such regularization. GJS encourages consistency in a simple way that exhibits other desirable properties for learning with noisy labels. Importantly, Jensen-Shannon-based consistency loss functions have been used to improve test-time robustness to image corruptions [6] and adversarial examples [17], which further verifies the general usefulness of GJS. In this work, we study such loss functions for a different goal: training-time label-noise robustness. In this context, our thorough analytical and empirical results are, to the best of our knowledge, novel.
+
+Recently, loss functions with information-theoretic motivations have been proposed [18, 19]. JS, with an apparent information-theoretic interpretation, has a strong connection to those. Especially, the latter is a close concurrent work studying JS and other divergences from the family of f-divergences [20]. However, in this work, we consider a generalization to more than two distributions and study the role of $\pi _ { 1 }$ , which they treat as a constant $\begin{array} { r } { \check { \boldsymbol { \pi } } _ { 1 } = \frac { 1 } { 2 } \boldsymbol { \cdot } } \end{array}$ ). These differences lead to improved performance and novel theoretical results, e.g., Proposition 1 and 2. Lastly, another generalization of JS was recently presented by Nielsen [21], where the arithmetic mean is generalized to abstract means.
+
+# 4 Experiments
+
+This section, first, empirically investigates the effectiveness of the proposed losses for learning with noisy labels on synthetic (Section 4.1) and real-world noise (Section 4.2). This is followed by several experiments and ablation studies (Section 4.3) to shed light on the properties of JS and GJS through empirical substantiation of the theories and claims provided in Section 2. All these additional experiments are done on the more challenging CIFAR-100 dataset.
+
+Table 1: Synthetic Noise Benchmark on CIFAR. We reimplement other noise-robust loss functions into the same learning setup and ResNet-34, including label smoothing (LS), Bootstrap (BS), Symmetric CE (SCE), Generalized CE (GCE), and Normalized CE $( \mathrm { N C E + R C E } )$ ). We used same hyperparameter optimization budget and mechanism for all the prior works and ours. Mean test accuracy and standard deviation are reported from five runs and the statistically-significant top performers are boldfaced. The thorough analysis is evident from the higher performance of CE in our setup compared to prior works. GJS achieves state-of-the-art results for different noise rates, types, and datasets. Generally, GJS’s efficacy is more evident for the more challenging CIFAR-100 dataset.   
+
+<table><tr><td>Dataset</td><td>Method</td><td>No Noise</td><td colspan="4">Symmetric Noise Rate</td><td colspan="2">Asymmetric Noise Rate</td></tr><tr><td></td><td></td><td>0%</td><td>20%</td><td>40%</td><td>60%</td><td>80%</td><td>20%</td><td>40%</td></tr><tr><td></td><td>CE</td><td>95.77 ± 0.11</td><td>91.63 ±0.27</td><td>87.74 ±0.46</td><td>81.99 ± 0.56</td><td>66.51 ± 1.49</td><td>92.77 ± 0.24</td><td>87.12 ± 1.21</td></tr><tr><td></td><td>BS</td><td>94.58± 0.25</td><td>91.68 ±0.32</td><td>89.23 ±0.16</td><td>82.65 ± 0.57</td><td>16.97 ± 6.36</td><td>93.06±0.25</td><td>88.87 ± 1.06</td></tr><tr><td></td><td>LS</td><td>95.64 ± 0.12</td><td>93.51 ± 0.20</td><td>89.90 ±0.20</td><td>83.96±0.58</td><td>67.35 ± 2.71</td><td>92.94 ± 0.17</td><td>88.10 ±0.50</td></tr><tr><td>CIFAR-10</td><td>SCE</td><td>95.75 ± 0.16</td><td>94.29 ± 0.14</td><td>92.72 ±0.25</td><td>89.26± 0.37</td><td>80.68 ± 0.42</td><td>93.48± 0.31</td><td>84.98±0.76</td></tr><tr><td></td><td>GCE</td><td>95.75 ± 0.14</td><td>94.24 ± 0.18</td><td>92.82 ± 0.11</td><td>89.37 ±0.27</td><td>79.19 ± 2.04</td><td>92.83 ±0.36</td><td>87.00 ±0.99</td></tr><tr><td></td><td>NCE+RCE</td><td>95.36±0.09</td><td>94.27 ± 0.18</td><td>92.03 ±0.31</td><td>87.30 ± 0.35</td><td>77.89 ± 0.61</td><td>93.87 ± 0.03</td><td>86.83 ±0.84</td></tr><tr><td></td><td>JS</td><td>95.89 ±0.10</td><td>94.52 ± 0.21</td><td>93.01 ±0.22</td><td>89.64 ±0.15</td><td>76.06 ±0.85</td><td>92.18 ± 0.31</td><td>87.99 ± 0.55</td></tr><tr><td></td><td>GJS</td><td>95.91 ± 0.09</td><td>95.33 ± 0.18</td><td>93.57 ± 0.16</td><td>91.64 ± 0.22</td><td>79.11 ± 0.31</td><td>93.94±0.25</td><td>89.65± 0.37</td></tr><tr><td></td><td>CE</td><td>77.60 ± 0.17</td><td>65.74 ± 0.22</td><td>55.77 ± 0.83</td><td>44.42 ± 0.84</td><td>10.74 ± 4.08</td><td>66.85±0.32</td><td>49.45 ± 0.37</td></tr><tr><td></td><td>BS</td><td>77.65± 0.29</td><td>72.92 ± 0.50</td><td>68.52 ± 0.54</td><td>53.80 ± 1.76</td><td>13.83 ± 4.41</td><td>73.79 ± 0.43</td><td>64.67 ± 0.69</td></tr><tr><td></td><td>LS</td><td>78.60 ±0.04</td><td>74.88 ± 0.15</td><td>68.41 ± 0.20</td><td>54.58 ± 0.47</td><td>26.98 ± 1.07</td><td>73.17 ± 0.46</td><td>57.20±0.85</td></tr><tr><td>CIFAR-100</td><td>SCE</td><td>78.29±0.24</td><td>74.21 ± 0.37</td><td>68.23 ±0.29</td><td>59.28 ± 0.58</td><td>26.80 ± 1.11</td><td>70.86 ± 0.44</td><td>51.12 ± 0.37</td></tr><tr><td></td><td>GCE</td><td>77.65 ± 0.17</td><td>75.02 ± 0.24</td><td>71.54 ± 0.39</td><td>65.21 ± 0.16</td><td>49.68 ±0.84</td><td>72.13 ±0.39</td><td>51.50 ± 0.71</td></tr><tr><td></td><td>NCE+RCE</td><td>74.66 ± 0.21</td><td>72.39± 0.24</td><td>68.79 ±0.29</td><td>62.18 ± 0.35</td><td>31.63 ± 3.59</td><td>71.35 ± 0.16</td><td>57.80±0.52</td></tr><tr><td></td><td>JS</td><td>77.95± 0.39</td><td>75.41 ± 0.28</td><td>71.12 ± 0.30</td><td>64.36 ±0.34</td><td>45.05 ± 0.93</td><td>71.70 ± 0.36</td><td>49.36±0.25</td></tr><tr><td></td><td>GJS</td><td>79.27 ±0.29</td><td>78.05±0.25</td><td>75.71±0.25</td><td>70.15 ±0.30</td><td>44.49 ± 0.53</td><td>74.60 ± 0.47</td><td>63.70 ±0.22</td></tr></table>
+
+Experimental Setup. We use ResNet 34 and 50 for experiments on CIFAR and WebVision datasets respectively and optimize them using SGD with momentum. The complete details of the training setup can be found in Appendix A. Most importantly, we take three main measures to ensure a fair and reliable comparison throughout the experiments: 1) we reimplement all the loss functions we compare with in a single shared learning setup, 2) we use the same hyperparameter optimization budget and mechanism for all the prior works and ours, and 3) we train and evaluate five networks for individual results, where in each run the synthetic noise, network initialization, and data-order are differently randomized. The thorough analysis is evident from the higher performance of CE in our setup compared to prior works. Where possible, we report mean and standard deviation and denote the statistically-significant top performers with student t-test.
+
+# 4.1 Synthetic Noise Benchmarks: CIFAR
+
+Here, we evaluate the proposed loss functions on the CIFAR datasets with two types of synthetic noise: symmetric and asymmetric. For symmetric noise, the labels are, with probability $\eta$ , resampled from a uniform distribution over all labels. For asymmetric noise, we follow the standard setup of Patrini et al. [22]. For CIFAR-10, the labels are modified, with probability $\eta$ , as follows: truck automobile, $b i r d $ airplane, $c a t  d o g$ , and $d e e r \to h o r s e$ . For CIFAR-100, labels are, with probability $\eta$ , cycled to the next sub-class of the same “super-class”, e.g. the labels of super-class “vehicles $1 ^ { \circ }$ are modified as follows: bicycle $ b u s  m o t o r c y c l e  p i c k u p t r u c k  t r a i n  b i c y c l e$ .
+
+We compare with other noise-robust loss functions such as label smoothing (LS) [23], Bootstrap (BS) [24], Symmetric Cross-Entropy (SCE) [4], Generalized Cross-Entropy (GCE) [3], and the $\mathrm { N C E + R C E }$ loss of Ma et al. [5]. Here, we do not compare to methods that propose a full pipeline since, first, a conclusive comparison would require re-implementation and individual evaluation of several components and second, robust loss functions can be considered orthogonal to them.
+
+Results. Table 1 shows the results for symmetric and asymmetric noise on CIFAR-10 and CIFAR100. GJS performs similarly or better than other methods for different noise rates, noise types, and data sets. Generally, GJS’s efficacy is more evident for the more challenging CIFAR-100 dataset. For example, on $60 \%$ uniform noise on CIFAR-100, the difference between GJS and the second best (GCE) is 4.94 percentage points, while our results on $80 \%$ noise is lower than GCE. We attribute this to the high sensitivity of the results to the hyperparameter settings in such a high-noise rate which are also generally unrealistic (WebVision has $\sim 2 0 \%$ ). The performance of JS is consistently similar to the top performance of the prior works across different noise rates, types and datasets. In Section 4.3, we substantiate the importance of the consistency term, identified in Proposition 2, when going from JS to GJS that helps with the learning dynamics and reduce the susceptibility to noise. In Appendix B.1, we provide results for GJS on instance-dependent synthetic noise [25]. Next, we test the proposed losses on a naturally-noisy dataset to see their efficacy in a real-world scenario.
+
+Table 2: Real-world Noise Benchmark on WebVision. Mean test accuracy and standard deviation from five runs are reported for the validation sets of (mini) WebVision and ILSVRC12. GJS with two networks correspond to the mean prediction of two independently trained GJS networks with different seeds for data augmentation and weight initialization. Here, GJS uses $Z = 1$ . Results marked with $\dagger$ are from Zheltonozhskii et al. [26].   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Architecture</td><td rowspan="2">Augmentation Networks</td><td rowspan="2"></td><td colspan="2">WebVision</td><td colspan="2">ILSVRC12</td></tr><tr><td>Top1</td><td>Top5</td><td>Top1</td><td>Top 5</td></tr><tr><td>ELR+ [27]†</td><td>Inception-ResNet-V2</td><td>Mixup</td><td>2</td><td>77.78</td><td>91.68</td><td>70.29</td><td>89.76</td></tr><tr><td>DivideMix [16]t]</td><td>Inception-ResNet-V2</td><td>Mixup</td><td>2</td><td>77.32</td><td>91.64</td><td>75.20</td><td>90.84</td></tr><tr><td>DivideMix [16]t</td><td>ResNet-50</td><td>Mixup</td><td>2</td><td>76.32 ± 0.36</td><td>90.65 ± 0.16</td><td>74.42 ± 0.29</td><td>91.21±0.12</td></tr><tr><td>CE</td><td>ResNet-50</td><td>ColorJitter</td><td>1</td><td>70.69 ± 0.66</td><td>88.64 ± 0.17</td><td>67.32 ± 0.57</td><td>88.00 ±0.49</td></tr><tr><td>JS</td><td>ResNet-50</td><td>ColorJitter</td><td>1</td><td>74.56± 0.32</td><td>91.09 ±0.08</td><td>70.36 ± 0.12</td><td>90.60 ±0.09</td></tr><tr><td>GJS</td><td>ResNet-50</td><td>ColorJitter</td><td>1</td><td>77.99 ± 0.35</td><td>90.62 ±0.28</td><td>74.33 ± 0.46</td><td>90.33±0.20</td></tr><tr><td>GJS</td><td>ResNet-50</td><td>ColorJitter</td><td>2</td><td>79.28±0.24</td><td>91.22 ± 0.30</td><td>75.50 ± 0.17</td><td>91.27±0.26</td></tr></table>
+
+# 4.2 Real-World Noise Benchmark: WebVision
+
+WebVision v1 is a large-scale image dataset collected by crawling Flickr and Google, which resulted in an estimated $20 \%$ of noisy labels [28]. There are 2.4 million images of the same thousand classes as ILSVRC12. Here, we use a smaller version called mini WebVision [29] consisting of the first 50 classes of the Google subset. We compare CE, JS, and GJS on WebVision following the same rigorous procedure as for the synthetic noise. However, upon request by the reviewers, we also compare with the reported results of some state-of-the-art elaborate techniques. This comparison deviates from our otherwise systematic analysis.
+
+Results. Table 2, as the common practice, reports the performances on the validation sets of WebVision and ILSVRC12 (first 50 classes). Both JS and GJS exhibit large margins with standard CE, especially for top-1 accuracy. Top-5 accuracy, due to its admissibility of wrong top predictions, can obscure the susceptibility to noise-fitting and thus indicates smaller but still significant improvements.
+
+The two state-of-the-art methods on this dataset were DivideMix [16] and $\mathrm { E L R + }$ [27]. Compared to our setup, both these methods use a stronger network (Inception-ResNet-V2 vs ResNet-50), stronger augmentations (Mixup vs color jittering) and co-train two networks. Furthermore, $\mathrm { E L R + }$ uses an exponential moving average of weights and DivideMix treats clean and noisy labeled examples differently after separating them using Gaussian mixture models. Despite these differences, GJS performs as good or better in terms of top-1 accuracy on WebVision and significantly outperforms $\mathrm { E L R + }$ on ILSVRC12 (70.29 vs 74.33). The importance of these differences becomes apparent as 1) the top-1 accuracy for DivideMix degrades when using ResNet-50, and 2) the performance of GJS improves by adding one of their components, i.e. the use of two networks. We train an ensemble of two independent networks with the GJS loss and average their predictions (last row of Table 2). This simple extension, which requires no change in the training code, gives significant improvements. To the best of our knowledge, this is the highest reported top-1 accuracy on WebVision and ILSVRC12 when no pre-training is used.
+
+In Appendix B.2, we show state-of-the-art results when using GJS on two other real-world noisy datasets: ANIMAL-10N [30] and Food-101N [31].
+
+So far, the experiments demonstrated the robustness of the proposed loss function (regarding Proposition 3) via the significant improvement of the final accuracy on noisy datasets. While this was central and informative, it is also important to investigate whether this improvement comes from the theoretical properties that were argued for JS and GJS. In what follows, we devise several such experiments, in an effort to substantiate the theoretical claims and conjectures.
+
+![](images/48861da0caca096c5aef4b7c6a2f45f415222a7bd0a52209f03c6370b8ac81e4.jpg)  
+Figure 4: Effect of $\pi _ { 1 }$ . Validation accuracy of JS and GJS during training with symmetric noise on CIFAR100. From Proposition 1, JS behaves like CE and MAE for low and high values of $\pi _ { 1 }$ , respectively. The signs of noise-fitting for $\pi _ { 1 } = 0 . 1$ on $60 \%$ noise (b), and slow learning of $\pi _ { 1 } = 0 . 9$ (a-b), show this in practice. The GJS loss does not exhibit overfitting for low values of $\pi _ { 1 }$ and learns quickly for large values of $\pi _ { 1 }$ (c-d).
+
+![](images/a67d759109b64075db34e39ff1fdc57bee0fbcaad6e7f81a5629a61ea0f843e6.jpg)  
+Figure 5: Effect of M. Validation accuracy for increasing number of distributions $( M )$ and different symmetric noise rates on CIFAR-100 with $\pi _ { 1 } = { \frac { 1 } { 2 } } \quad$ . For all noise rates, using three instead of two distributions results in a higher accuracy. Going beyond three distributions is only helpful for lower noise rates. For simplicity we use $M = 3$ (corresponding to two augmentations) for all of our experiments.
+
+# 4.3 Towards a Better Understanding of the Jensen-Shannon-based Loss Functions
+
+Here, we study the behavior of the losses for different distribution weights $\pi _ { 1 }$ , number of distributions $M$ , and epochs. We also provide insights on why GJS performs better than JS.
+
+How does $\pi _ { 1 }$ control the trade-off of robustness and learnability? In Figure 4, we plot the validation accuracy during training for both JS and GJS at different values of $\pi _ { 1 }$ and noise rates $\eta$ From Proposition 1, we expect JS to behave as CE for low values of $\pi _ { 1 }$ and as MAE for larger values of $\pi _ { 1 }$ . Figure 4 (a-b) confirms this. Specifically, $\pi _ { 1 } = 0 . 1$ learns quickly and performs well for low noise but overfits for $\eta = 0 . 6$ (characteristic of non-robust CE), on the other hand, $\pi _ { 1 } = 0 . 9$ learns slowly but is robust to high noise rates (characteristic of noise-robust MAE).
+
+In Figure 4 (c-d), we observe three qualitative improvements of GJS over JS: 1) no signs of overfitting to noise for large noise rates with low values of $\pi _ { 1 }$ , 2) better learning dynamics for large values of $\pi _ { 1 }$ that otherwise learns slowly, and 3) converges to a higher validation accuracy.
+
+How many distributions to use? Figure 5 depicts validation accuracy for varying number of distributions $M$ . For all noise rates, we observe a performance increase going from $M = 2$ to $M = 3$ However, the performance of $M > 3$ depends on the noise rate. For lower noise rates, having more than three distributions can improve the performance. For higher noise rates e.g. $6 0 \%$ , having $M > 3$ degrades the performance. We hypothesise this is due to: 1) at high noise rates, there are only a few correctly labeled examples that can help guide the learning, and 2) going from $M = 2$ to $M = 3$ adds a consistency term, while $M > 3$ increases the importance of the consistency term in Proposition 2. Therefore, for a large enough M, the loss will find it easier to keep the consistency term low (keep predictions close to uniform as at the initialization), instead of generalizing based on the few clean examples. For simplicity, we have used $M = 3$ for all experiments with GJS.
+
+Is the improvements of GJS over JS due to mean prediction or consistency? Proposition 2 decomposed GJS into a JS term with a mean prediction $( { \bar { p } } _ { > 1 } )$ and a consistency term operating on all distributions but the target. In Table 3, we compare the performance of JS and GJS to GJS without the consistency term, i.e., $\mathcal { L } _ { \mathrm { J S } _ { \pi ^ { \prime } } } ( e ^ { ( y ) } , \bar { p } _ { > 1 } )$ . The results suggest that the improvement of GJS over JS can be attributed to the consistency term.
+
+Figure 4 (a-b) showed that JS improves the learning dynamics of MAE by blending it with CE, controlled by $\pi _ { 1 }$ . Similarly, we see here that the consistency term also improves the learning dynamics (underfitting and convergence speed) of MAE. Interestingly, Figure 4 (c-d), shows the higher values of $\pi _ { 1 }$ (closer to MAE) work best for GJS, hinting that, the consistency term improves the learning dynamics of MAE so much so that the role of CE becomes less important.
+
+Table 3: Effect of Consistency. Validation accuracy for JS, GJS w/o the consistency term in Proposition 2, and GJS for $4 0 \%$ noise on the CIFAR-100 dataset. Using the mean of two predictions in the JS loss does not improve performance. On the other hand, adding the consistency term significantly helps.   
+
+<table><tr><td>Method</td><td>Accuracy</td></tr><tr><td>LJs(e(）,p（2））</td><td>71.0</td></tr><tr><td>LJsπ,(e(）,p&gt;1）</td><td>68.7</td></tr><tr><td>LGJs(e(y）,p(2）,p(3）)</td><td>74.3</td></tr></table>
+
+Table 4: Effect of GJS. Validation accuracy when using different loss functions for clean and noisy examples of the CIFAR-100 training set with $40 \%$ symmetric noise. Noisy examples benefit significantly more from GJS than clean examples (74.1 vs 72.9).   
+
+<table><tr><td colspan="2">Method</td><td colspan="3">T1</td></tr><tr><td>Clean</td><td>Noisy</td><td>0.1</td><td>0.5</td><td>0.9</td></tr><tr><td>JS</td><td>JS</td><td>70.0</td><td>71.5</td><td>55.3</td></tr><tr><td>GJS</td><td>JS</td><td>72.6</td><td>72.9</td><td>70.2</td></tr><tr><td>JS</td><td>GJS</td><td>71.0</td><td>74.1</td><td>68.0</td></tr><tr><td>GJS</td><td>GJS</td><td>71.3</td><td>74.7</td><td>73.8</td></tr></table>
+
+Is GJS mostly helping the clean or noisy examples? To better understand the improvements of GJS over JS, we perform an ablation with different losses for clean and noisy examples, see Table 4. We observe that using GJS instead of JS improves performance in all cases. Importantly, using GJS only for the noisy examples performs significantly better than only using it for the clean examples (74.1 vs 72.9). The best result is achieved when using GJS for both clean and noisy examples but still close to the noisy-only case (74.7 vs 74.1).
+
+How is different choices of perturbations affecting GJS? In this work, we use stochastic augmentations for $\mathcal { A }$ , see Appendix A.1 for details. Table 5 reports validation results on $40 \%$ symmetric and asymmetric noise on CIFAR-100 for varying types of augmentation. We observe that all methods improve their performance with stronger augmentation and that GJS achieves the best results in all cases. Also, note that we use weak augmentation for all naturally-noisy datasets (WebVision, ANIMAL-10N, and Food-101N) and still get state-of-the-art results.
+
+How fast is the convergence? We found that some baselines (especially the robust $\mathrm { N C E + R C E }$ had slow convergence. Therefore, we used 400 epochs for all methods to make sure all had time to converge properly. Table 6 shows results on $40 \%$ symmetric and asymmetric noise on CIFAR-100 when the number of epochs has been reduced by half.
+
+Is training with the proposed losses leading to more consistent networks? Our motivation for investigating losses based on Jensen-Shannon divergence was partly due to the observation in Figure 1 that consistency and accuracy correlate when learning with CE loss. In Figure 6, we compare CE, JS, and GJS losses in terms of validation accuracy and consistency during training on CIFAR-100 with $40 \%$ symmetric noise. We find that the networks trained with JS and GJS losses are more consistent and has higher accuracy. In Appendix B.7, we report the consistency of the networks in Table 1.
+
+Summary of experiments in the appendix. Due to space limitations, we report several important experiments in the appendix. We evaluate the effectiveness of GJS on 1) instance-dependent synthetic noise (Section B.1), and 2) real-world noisy datasets ANIMAL-10N and Food-101N (Section B.2). We also investigate the importance of 1) losses being symmetric and bounded for learning with noisy labels (Section B.3), and 2) a clean vs noisy validation set for hyperparameter selection and the effect of a single set of parameters for all noise rates (Section B.5).
+
+Table 5: Effect of Augmentation Strategy. Validation accuracy for training w/o CutOut(-CO) or w/o RandAug(-RA) or w/o both(weak) on $40 \%$ symmetric and asymmetric noise on CIFAR-100. All methods improves by stronger augmentations. GJS performs best for all types of augmentations.   
+
+<table><tr><td rowspan="3">Method</td><td colspan="4">Symmetric</td><td colspan="4">Asymmetric</td></tr><tr><td>Full</td><td>-CO</td><td>-RA</td><td>Weak Full</td><td></td><td>-C0</td><td>-RA</td><td>Weak</td></tr><tr><td>GCE</td><td>70.8</td><td>64.2</td><td>64.1</td><td>58.0</td><td>51.7</td><td>44.9</td><td>46.6</td><td>42.9</td></tr><tr><td>NCE+RCE</td><td>68.5</td><td>66.6</td><td>68.3</td><td>61.7</td><td>57.5</td><td>52.1</td><td>49.5</td><td>44.4</td></tr><tr><td>GJS</td><td>74.8</td><td>71.3</td><td>70.6</td><td>66.5</td><td>62.6</td><td>56.8</td><td>52.2</td><td>44.9</td></tr></table>
+
+Table 6: Effect of Number of Epochs. Validation accuracy for training with 200 and 400 epochs for $40 \%$ symmetric and asymmetric noise on CIFAR-100. GJS still outperforms the baselines and $\mathrm { N C E + R C E }$ ’s performance is reduced heavily by the decrease in epochs.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Symmetric</td><td colspan="2">Asymmetric</td></tr><tr><td>200</td><td>400</td><td>200</td><td>400</td></tr><tr><td>GCE</td><td>70.3</td><td>70.8</td><td>39.1</td><td>51.7</td></tr><tr><td>NCE+RCE</td><td>60.0</td><td>68.5</td><td>35.0</td><td>57.5</td></tr><tr><td>GJS</td><td>72.9</td><td>74.8</td><td>43.2</td><td>62.6</td></tr></table>
+
+![](images/640df0ee49a168a2347bf5044f3ecb4712f30f38c30acf34b7bef3fac730e798.jpg)  
+Figure 6: Evolution of a trained network’s consistency for the CE, JS, and GJS losses. We plot the evolution of the validation accuracy (a) and network’s consistency on clean (b) and noisy (c) examples of the training set of CIFAR-100 when learning with $40 \%$ symmetric noise. All losses use the same learning rate and weight decay and both JS and GJS use $\pi _ { 1 } = 0 . 5$ . The consistency of the learnt function and the accuracy closely correlate. The accuracy and consistency of JS and GJS improve during training, while both degrade when learning with CE loss.
+
+# 5 Limitations & Future Directions
+
+We empirically showed that the consistency of the network around noisy data degrades as it fits noise and accordingly proposed a loss based on generalized Jensen-Shannon divergence (GJS). While we empirically verified the significant role of consistency regularization in robustness to noise, we only theoretically showed the robustness $B _ { L } = B _ { U , }$ ) of GJS at its limit $\pi _ { 1 }  1 $ ) where the consistency term gradually vanishes. Therefore, the main limitation is the lack of a theoretical proof of the robustness of the consistency term in Proposition 2. This is, in general, an important but understudied area, also for the literature of self- or semi-supervised learning and thus is of utmost importance for future works.
+
+Secondly, we had an important observation that GJS with $M > 3$ might not perform well under high noise rates. While we have some initial conjectures, this phenomenon deserves a systematic analysis both empirically and theoretically.
+
+Finally, a minor practical limitation is the added computations for GJS forward passes, however this applies to training time only and in all our experiments, we only use one extra prediction ( $M = 3$ ).
+
+# 6 Final Remarks
+
+We first made two central observations that (i) robust loss functions have an underfitting issue and (ii) consistency of noise-fitting networks is significantly lower around noisy data points. Correspondingly, we proposed two loss functions, JS and GJS, based on Jensen-Shannon divergence that (i) interpolates between noise-robust MAE and fast-converging CE, and (ii) encourages consistency around training data points. This simple proposal led to state-of-the-art performance on both synthetic and real-world noise datasets even when compared to the more elaborate pipelines such as DivideMix or $\mathrm { E L R + }$ . Furthermore, we discussed their robustness within the theoretical construction of Ghosh et al. [2]. By drawing further connections to other seminal loss functions such as CE, MAE, GCE, and consistency regularization, we uncovered other desirable or informative properties. We further empirically studied different aspects of the losses that corroborate various theoretical properties.
+
+Overall, we believe the paper provides informative theoretical and empirical evidence for the usefulness of two simple and novel JS divergence-based loss functions for learning under noisy data that achieve state-of-the-art results. At the same time, it opens interesting future directions.
+
+Ethical Considerations. Considerable resources are needed to create labeled data sets due to the burden of manual labeling process. Thus, the creators of large annotated datasets are mostly limited to well-funded companies and academic institutions. In that sense, developing robust methods against label noise enables less affluent organizations or individuals to benefit from labeled datasets since imperfect or automatic labeling can be used instead. On the other hand, proliferation of such harvested datasets can increase privacy concerns arising from redistribution and malicious use.
+
+Acknowledgement. This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
+
+References   
+[1] Lucas Beyer, Olivier J Hénaff, Alexander Kolesnikov, Xiaohua Zhai, and Aäron van den Oord. Are we done with imagenet? arXiv preprint arXiv:2006.07159, 2020.   
+[2] Aritra Ghosh, Himanshu Kumar, and PS Sastry. Robust loss functions under label noise for deep neural networks. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, pages 1919–1925, 2017.   
+[3] Zhilu Zhang and Mert Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. In Advances in neural information processing systems, pages 8778–8788, 2018.   
+[4] Yisen Wang, Xingjun Ma, Zaiyi Chen, Yuan Luo, Jinfeng Yi, and James Bailey. Symmetric cross entropy for robust learning with noisy labels. In Proceedings of the IEEE International Conference on Computer Vision, pages 322–330, 2019.   
+[5] Xingjun Ma, Hanxun Huang, Yisen Wang, Simone Romano, Sarah Erfani, and James Bailey. Normalized loss functions for deep learning with noisy labels, 2020.   
+[6] Dan Hendrycks, Norman Mu, Ekin D. Cubuk, Barret Zoph, Justin Gilmer, and Balaji Lakshminarayanan. Augmix: A simple data processing method to improve robustness and uncertainty. In International Conference on Learning Representation, 2020.   
+[7] Avital Oliver, Augustus Odena, Colin Raffel, Ekin D Cubuk, and Ian J Goodfellow. Realistic evaluation of deep semi-supervised learning algorithms. arXiv preprint arXiv:1804.09170, 2018.   
+[8] Jianhua Lin. Divergence measures based on the shannon entropy. IEEE Transactions on Information theory, 37(1):145–151, 1991.   
+[9] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization, 2017.   
+[10] Yang Liu and Hongyi Guo. Peer loss functions: Learning from noisy labels without knowing noise rates. In Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 6226–6236. PMLR, 13–18 Jul 2020.   
+[11] Nagarajan Natarajan, Inderjit S Dhillon, Pradeep K Ravikumar, and Ambuj Tewari. Learning with noisy labels. In Advances in neural information processing systems, pages 1196–1204, 2013.   
+[12] Sainbayar Sukhbaatar, Joan Bruna, Manohar Paluri, Lubomir Bourdev, and Rob Fergus. Training convolutional networks with noisy labels. In Proceedings of the international conference on learning representation, 2015.   
+[13] Giorgio Patrini, Alessandro Rozza, Aditya Krishna Menon, Richard Nock, and Lizhen Qu. Making deep neural networks robust to label noise: A loss correction approach. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1944–1952, 2017.   
+[14] Bo Han, Jiangchao Yao, Gang Niu, Mingyuan Zhou, Ivor Tsang, Ya Zhang, and Masashi Sugiyama. Masking: A new perspective of noisy supervision. In Advances in Neural Information Processing Systems, pages 5836–5846, 2018.   
+[15] Xiaobo Xia, Tongliang Liu, Nannan Wang, Bo Han, Chen Gong, Gang Niu, and Masashi Sugiyama. Are anchor points really indispensable in label-noise learning? In Advances in Neural Information Processing Systems, pages 6838–6849, 2019.   
+[16] Junnan Li, Richard Socher, and Steven CH Hoi. Dividemix: Learning with noisy labels as semi-supervised learning. In International Conference on Learning Representation, 2020.   
+[17] Jihoon Tack, Sihyun Yu, Jongheon Jeong, Minseon Kim, Sung Ju Hwang, and Jinwoo Shin. Consistency regularization for adversarial robustness, 2021.   
+[18] Yilun Xu, Peng Cao, Yuqing Kong, and Yizhou Wang. L_dmi: A novel information-theoretic loss function for training deep nets robust to label noise. In Advances in Neural Information Processing Systems, pages 6225–6236, 2019.   
+[19] Jiaheng Wei and Yang Liu. When optimizing f-divergence is robust with label noise. In International Conference on Learning Representation, 2021.   
+[20] I. CSISZAR. Information-type measures of difference of probability distributions and indirect observation. Studia Scientiarum Mathematicarum Hungarica, 2:229–318, 1967.   
+[21] Frank Nielsen. On the jensen–shannon symmetrization of distances relying on abstract means. Entropy, 21(5), 2019.   
+[22] Giorgio Patrini, Alessandro Rozza, Aditya Menon, Richard Nock, and Lizhen Qu. Making deep neural networks robust to label noise: a loss correction approach, 2017.   
+[23] Michal Lukasik, Srinadh Bhojanapalli, Aditya Krishna Menon, and Sanjiv Kumar. Does label smoothing mitigate label noise? In International Conference on Machine Learning, 2020.   
+[24] Scott Reed, Honglak Lee, Dragomir Anguelov, Christian Szegedy, Dumitru Erhan, and Andrew Rabinovich. Training deep neural networks on noisy labels with bootstrapping. arXiv preprint arXiv:1412.6596, 2014.   
+[25] Yikai Zhang, Songzhu Zheng, Pengxiang Wu, Mayank Goswami, and Chao Chen. Learning with feature-dependent label noise: A progressive approach, 2021.   
+[26] Evgenii Zheltonozhskii, Chaim Baskin, Avi Mendelson, Alex M. Bronstein, and Or Litany. Contrast to divide: Self-supervised pre-training for learning with noisy labels, 2021.   
+[27] Sheng Liu, Jonathan Niles-Weed, Narges Razavian, and Carlos Fernandez-Granda. Earlylearning regularization prevents memorization of noisy labels, 2020.   
+[28] Wen Li, Limin Wang, Wei Li, Eirikur Agustsson, and Luc Van Gool. Webvision database: Visual learning and understanding from web data, 2017.   
+[29] Lu Jiang, Zhenyuan Zhou, Thomas Leung, Jia Li, and Fei-Fei Li. Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels. In ICML, 2018.   
+[30] Hwanjun Song, Minseok Kim, and Jae-Gil Lee. SELFIE: Refurbishing unclean samples for robust deep learning. In ICML, 2019.   
+[31] Kuang-Huei Lee, Xiaodong He, Lei Zhang, and Linjun Yang. Cleannet: Transfer learning for scalable image classifier training with label noise. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.   
+[32] Ekin D. Cubuk, Barret Zoph, Jonathon Shlens, and Quoc V. Le. Randaugment: Practical automated data augmentation with a reduced search space, 2019.   
+[33] Terrance DeVries and Graham W. Taylor. Improved regularization of convolutional neural networks with cutout, 2017.   
+[34] Junnan Li, Yongkang Wong, Qi Zhao, and Mohan Kankanhalli. Learning to learn from noisy labeled data, 2019.   
+[35] Duc Tam Nguyen, Chaithanya Kumar Mummadi, Thi Phuong Nhung Ngo, Thi Hoai Phuong Nguyen, Laura Beggel, and Thomas Brox. Self: Learning to filter noisy labels with selfensembling. In International Conference on Learning Representation, 2019.   
+[36] Curtis G Northcutt, Tailin Wu, and Isaac L Chuang. Learning with confident examples: Rank pruning for robust classification with noisy labels. arXiv preprint arXiv:1705.01936, 2017.   
+[37] Daiki Tanaka, Daiki Ikami, Toshihiko Yamasaki, and Kiyoharu Aizawa. Joint optimization framework for learning with noisy labels. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5552–5560, 2018.   
+[38] Arash Vahdat. Toward robustness against label noise in training deep discriminative neural networks. In Advances in Neural Information Processing Systems, pages 5596–5605, 2017.   
+[39] Ahmet Iscen, Giorgos Tolias, Yannis Avrithis, Ondrej Chum, and Cordelia Schmid. Graph convolutional networks for learning with few clean and many noisy labels. In Proceedings of the European Conference on Computer Vision, 2020.   
+[40] Paul Hongsuck Seo, Geeho Kim, and Bohyung Han. Combinatorial inference against label noise. In Advances in Neural Information Processing Systems, pages 1173–1183, 2019.   
+[41] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2818–2826, 2016.   
+[42] Takeru Miyato, Shin-ichi Maeda, Masanori Koyama, and Shin Ishii. Virtual adversarial training: a regularization method for supervised and semi-supervised learning. IEEE transactions on pattern analysis and machine intelligence, 41(8):1979–1993, 2018.   
+[43] David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin A Raffel. Mixmatch: A holistic approach to semi-supervised learning. In Advances in Neural Information Processing Systems, pages 5049–5059, 2019.   
+[44] Antti Tarvainen and Harri Valpola. Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. In Advances in neural information processing systems, pages 1195–1204, 2017.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] See Section C in the Appendix.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See footnote on the first page.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section A in the Appendix.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See main experiments in Table 1 & 2.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section A.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]
+
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/TiwPYwg3IRf/TiwPYwg3IRf_content_list.json

@@ -0,0 +1,1438 @@
+[
+    {
+        "type": "text",
+        "text": "Generalized Jensen-Shannon Divergence Loss for Learning with Noisy Labels ",
+        "text_level": 1,
+        "bbox": [
+            222,
+            122,
+            776,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Erik Englesson KTH Stockholm, Sweden engless@kth.se ",
+        "bbox": [
+            299,
+            226,
+            433,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hossein Azizpour KTH Stockholm, Sweden azizpour@kth.se ",
+        "bbox": [
+            566,
+            226,
+            697,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
+            535,
+            334
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Prior works have found it beneficial to combine provably noise-robust loss functions e.g., mean absolute error (MAE) with standard categorical loss function e.g. cross entropy (CE) to improve their learnability. Here, we propose to use Jensen-Shannon divergence as a noise-robust loss function and show that it interestingly interpolate between CE and MAE with a controllable mixing parameter. Furthermore, we make a crucial observation that CE exhibits lower consistency around noisy data points. Based on this observation, we adopt a generalized version of the JensenShannon divergence for multiple distributions to encourage consistency around data points. Using this loss function, we show state-of-the-art results on both synthetic (CIFAR), and real-world (e.g. WebVision) noise with varying noise rates. ",
+        "bbox": [
+            233,
+            347,
+            766,
+            486
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            501,
+            310,
+            518
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Labeled datasets, even the systematically annotated ones, contain noisy labels [1]. Therefore, designing noise-robust learning algorithms are crucial for the real-world tasks. An important avenue to tackle noisy labels is to devise noise-robust loss functions [2, 3, 4, 5]. Similarly, in this work, we propose two new noise-robust loss functions based on two central observations as follows. ",
+        "bbox": [
+            176,
+            531,
+            825,
+            587
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Observation I: Provably-robust loss functions can underfit the training data [2, 3, 4, 5].   \nObservation II: Standard networks show low consistency around noisy data points 1, see Figure 1. ",
+        "bbox": [
+            173,
+            593,
+            805,
+            622
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We first propose to use Jensen-Shannon divergence (JS) as a loss function, which we crucially show interpolates between the noise-robust mean absolute error (MAE) and the cross entropy (CE) that better fits the data through faster convergence. Figure 2 illustrates the CE-MAE interpolation. Regarding Observation II, we adopt the generalized version of Jensen-Shannon divergence (GJS) to encourage predictions on perturbed inputs to be consistent, see Figure 3. Notably, Jensen-Shannon divergence has previously shown promise for test-time robustness to domain shift [6], here we further argue for its training-time robustness to label noise. The key contributions of this work2 are: ",
+        "bbox": [
+            173,
+            627,
+            825,
+            726
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• We make a novel observation that a network predictions’ consistency is reduced for noisylabeled data when overfitting to noise, which motivates the use of consistency regularization. • We propose using Jensen-Shannon divergence (JS) and its multi-distribution generalization (GJS) as loss functions for learning with noisy labels. We relate JS to loss functions that are based on the noise-robustness theory of Ghosh et al. [2]. In particular, we prove that JS generalizes CE and MAE. Furthermore, we prove that GJS generalizes JS by incorporating consistency regularization in a single principled loss function. • We provide an extensive set of empirical evidences on several datasets, noise types and rates. They show state-of-the-art results and give in-depth studies of the proposed losses. ",
+        "bbox": [
+            212,
+            736,
+            825,
+            869
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/7c9ce0e5ce3b6ce01768b4299d109e57f0d618594199d5247062bc1e2f73ea34.jpg",
+        "image_caption": [
+            "Figure 1: Evolution of a trained network’s consistency as it overfits to noise using CE loss. Here we plot the evolution of the validation accuracy (a) and network’s consistency (as measured by GJS) on clean (b) and noisy (c) examples of the training set of CIFAR-100 for varying symmetric noise rates when learning with the cross-entropy loss. The consistency of the learnt function and the accuracy closely correlate. This suggests that enforcing consistency may help avoid fitting to noise. Furthermore, the consistency is degraded more significantly for the noisy data points. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            88,
+            812,
+            215
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Generalized Jensen-Shannon Divergence ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            324,
+            545,
+            342
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We propose two loss functions, the Jensen-Shannon divergence (JS) and its multi-distribution generalization (GJS). In this section, we first provide background and two observations that motivate our proposed loss functions. This is followed by definition of the losses, and then we show that JS generalizes CE and MAE similarly to other robust loss functions. Finally, we show how GJS generalizes JS to incorporate consistency regularization into a single principled loss function. We provide proofs of all theorems, propositions, and remarks in this section in Appendix C. ",
+        "bbox": [
+            173,
+            348,
+            825,
+            433
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 Background & Motivation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            449,
+            397,
+            463
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Supervised Classification. Assume a general function class3 $\\mathcal { F }$ where each $f \\in \\mathcal F$ maps an input $\\textbf { \\em x } \\in \\mathrm { ~ \\mathbb ~ X ~ }$ to the probability simplex $\\Delta ^ { K - 1 }$ , i.e. to a categorical distribution over $K$ classes $\\boldsymbol { y } \\in \\mathbb { Y } = \\{ 1 , 2 , \\dots , K \\}$ . We seek $f ^ { * } \\in { \\mathcal { F } }$ that minimizes a risk $R _ { \\mathcal { L } } ( f ) = \\mathbb { E } _ { \\mathcal { D } } [ \\mathcal { L } ( e ^ { ( y ) } , f ( \\pmb { x } ) ) ]$ , for some loss function $\\mathcal { L }$ and joint distribution $\\mathcal { D }$ over $\\mathbb { X } \\times \\mathbb { Y }$ , where $e ^ { ( y ) }$ is a $K$ -vector with one at index and zero elsewhere. In practice, $\\mathcal { D }$ is unknown and, instead, we use ${ \\cal S } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N }$ which are independently sampled from $\\mathcal { D }$ to minimize an empirical risk $\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathcal { L } ( e ^ { ( y _ { i } ) } , f ( \\pmb { x } _ { i } ) ) } \\end{array}$ =1. ",
+        "bbox": [
+            173,
+            474,
+            825,
+            565
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Learning with Noisy Labels. In this work, the goal is to learn from a noisy training distribution $\\mathcal { D } _ { \\eta }$ where the labels are changed, with probability $\\eta$ , from their true distribution $\\mathcal { D }$ . The noise is called instance-dependent if it depends on the input, asymmetric if it dependents on the true label, and symmetric if it is independent of both $_ { \\textbf { \\em x } }$ and $y$ . Let $f _ { \\eta } ^ { * }$ be the optimizer of the noisy distribution risk $R _ { \\mathcal { L } } ^ { \\eta } ( f )$ . A loss function $\\mathcal { L }$ is then called robust if $f _ { \\eta } ^ { * }$ also minimizes $R _ { \\mathcal { L } }$ . The MAE loss $( \\mathcal L _ { M A E } ( e ^ { ( y ) } , f ( \\pmb x ) ) : = \\| e ^ { ( y ) } - f ( \\pmb x ) \\| _ { 1 } )$ is robust but not CE [2]. ",
+        "bbox": [
+            173,
+            570,
+            825,
+            660
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Issue of Underfitting. Several works propose such robust loss functions and demonstrate their efficacy in preventing noise fitting [2, 3, 4, 5]. However, all those works have observed slow convergence of such robust loss functions leading to underfitting. This can be contrasted with CE that has fast convergence but overfits to noise. Ghosh et al. [2] mentions slow convergence of MAE and GCE [3] extensively analyzes the undefitting thereof. SCE [4] reports similar problems for the reverse cross entropy and proposes a linear combination with CE. Finally, Ma et al. [5] observe the same problem and consider a combination of “active” and “passive” loss functions. ",
+        "bbox": [
+            174,
+            665,
+            825,
+            763
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Consistency Regularization. This encourages a network to have consistent predictions for different perturbations of the same image, which has mainly been used for semi-supervised learning [7]. ",
+        "bbox": [
+            173,
+            768,
+            823,
+            797
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Motivation. In Figure 1, we show the validation accuracy and a measure of consistency during training with the CE loss for varying amounts of noise. First, we note that training with CE loss eventually overfits to noisy labels. Figure 1a, indicates that the higher the noise rate, the more accuracy drop when it starts to overfit to noise. Figure 1(b-c) shows the consistency of predictions for correct and noisy labeled examples of the training set, with the consistency measured as the ratio of examples that have the same class prediction for two perturbations of the same image, see ",
+        "bbox": [
+            174,
+            803,
+            825,
+            887
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/06fcb0a20948d227fbd1a3afea7ab398020ba3f9dd73b458205a3ff9e6a6822d.jpg",
+        "image_caption": [
+            "Figure 2: JS loss generalizes CE and MAE. The Jensen-Shannon loss $( \\mathcal { L } _ { \\mathrm { J S } } )$ for different values of the hyperparameter $\\pi _ { 1 }$ . The JS loss interpolates between CE and MAE. For low values of $\\pi _ { 1 }$ , ${ \\mathcal { L } } _ { \\mathrm { J S } }$ behaves like CE and for increasing values of $\\pi _ { 1 }$ it behaves more like the noise robust MAE loss. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            88,
+            398,
+            190
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/5e65c0bbb5a37ec95fdbf5a65f0b12ec9dae92e8dd123a385e8f603049989121.jpg",
+        "image_caption": [
+            "Figure 3: GJS Dissection for $\\mathbf { M } = \\mathbf { K } = 3 \\colon$ The decomposition of ${ \\mathcal { L } } _ { \\mathrm { G J S } }$ (left) into a JS term (middle) and a consistency term (right) from Proposition 2. Each point in the simplex correspond to a $\\pmb { p } ^ { ( 3 ) } \\in \\Delta ^ { 2 }$ , where the color represents the value of the loss at that point. It can be seen that there are two ways to minimize ${ \\mathcal { L } } _ { \\mathrm { G J S } }$ , either by making the predictions similar to the label (middle) or similar to the other predictions (right) to increase consistency. To better highlight the variations of the losses, each loss has its own range of values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            437,
+            88,
+            815,
+            190
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Appendix B.6 for more details. A clear correlation is observed between the accuracy and consistency of the noisy examples. This suggests that maximizing consistency of predictions may improve the robustness to noise. Next, we define simple loss functions that (i) encourage consistency around data points and (ii) alleviate the “issue of underfitting” by interpolating between CE and MAE. ",
+        "bbox": [
+            173,
+            344,
+            825,
+            401
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Definitions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            412,
+            289,
+            428
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$D _ { \\mathbf { J } \\mathbf { S } }$ . Let $\\pmb { p } ^ { ( 1 ) } , \\pmb { p } ^ { ( 2 ) } \\in \\Delta ^ { K - 1 }$ have corresponding weights $\\pmb { \\pi } = [ \\pi _ { 1 } , \\pi _ { 2 } ] ^ { T } \\in \\Delta$ . Then, the JensenShannon divergence between $ { \\mathbf { \\mathit { p } } } ^ { ( 1 ) }$ and $ { p ^ { ( 2 ) } }$ is ",
+        "bbox": [
+            176,
+            435,
+            823,
+            468
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$D _ { 1 \\mathrm { S } _ { \\pi } } ( p ^ { ( 1 ) } , p ^ { ( 2 ) } ) : = H ( m ) - \\pi _ { 1 } H ( p ^ { ( 1 ) } ) - \\pi _ { 2 } H ( p ^ { ( 2 ) } ) = \\pi _ { 1 } D _ { \\mathrm { K L } } ( p ^ { ( 1 ) } | | m ) + \\pi _ { 2 } D _ { \\mathrm { K L } } ( p ^ { ( 2 ) } | | m )$ (1) with $H$ the Shannon entropy, and $\\pmb { m } ~ = ~ \\pi _ { 1 } \\pmb { p } ^ { ( 1 ) } + \\pi _ { 2 } \\pmb { p } ^ { ( 2 ) }$ . Unlike Kullback–Leibler divergence $( D _ { \\mathrm { K L } } ( \\pmb { p } ^ { ( 1 ) } \\| \\pmb { p } ^ { ( 2 ) } ) )$ or cross entropy (CE), JS is symmetric, bounded, does not require absolute continuity, and has a crucial weighting mechanism $( \\pi )$ , as we will see later. ",
+        "bbox": [
+            169,
+            481,
+            825,
+            549
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$D _ { \\mathbf { G J S } }$ . Similar to $D _ { \\mathrm { K L } }$ , $D _ { \\mathrm { J S } }$ satisfies $D _ { \\mathrm { J S } _ { \\pi } } ( { p } ^ { ( 1 ) } , { p } ^ { ( 2 ) } ) \\geq 0$ , with equality iff $\\pmb { p } ^ { ( 1 ) } = \\pmb { p } ^ { ( 2 ) }$ . For $D _ { \\mathrm { J S } }$ this is derived from Jensen’s inequality for the concave Shannon entropy. This property holds for finite number of distributions and motivates a generalization of $D _ { \\mathrm { J S } }$ to multiple distributions [8]: ",
+        "bbox": [
+            174,
+            553,
+            825,
+            598
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d1de297c7b0bc48ffef20f4b6c6f3001f219ce5bb9e5128beba209cef3b58c3a.jpg",
+        "text": "$$\nD _ { \\mathrm { G J S } _ { \\pi } } ( p ^ { ( 1 ) } , \\dots , p ^ { ( M ) } ) : = H { \\Bigl ( } \\sum _ { i = 1 } ^ { M } \\pi _ { i } p ^ { ( i ) } { \\Bigr ) } - \\sum _ { i = 1 } ^ { M } \\pi _ { i } H ( p ^ { ( i ) } ) = \\sum _ { i = 1 } ^ { M } \\pi _ { i } D _ { \\mathrm { K L } } { \\Bigl ( } p ^ { ( i ) } { \\Bigr \\| } \\sum _ { j = 1 } ^ { M } \\pi _ { j } p ^ { ( j ) } { \\Bigr ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            169,
+            602,
+            799,
+            646
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $M$ is the number of distributions, and $\\pmb { \\pi } = [ \\pi _ { 1 } , \\ldots , \\pi _ { M } ] ^ { T } \\in \\Delta ^ { M - 1 }$ ",
+        "bbox": [
+            174,
+            648,
+            666,
+            665
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Loss functions. We aim to use $D _ { \\mathrm { J S } }$ and $D _ { \\mathrm { G J S } }$ divergences, to measure deviation of the predictive distribution(s), $f ( { \\pmb x } )$ , from the target distribution, $e ^ { ( y ) }$ . Without loss of generality, hereafter, we dedicate $ { \\mathbf { \\mathit { p } } } ^ { ( 1 ) }$ to denote the target distribution. JS loss, therefore, can take the form of $D _ { \\mathrm { J S } _ { \\pi } } ( e ^ { ( y ) } , f ( { \\pmb x } ) )$ . Generalized JS loss is a less straight-forward construction since $D _ { \\mathrm { G J S } }$ can accommodate more predictive distributions. While various choices can be made for these distributions, in this work, we consider predictions associated with different random perturbations of a sample, denoted by $\\scriptstyle A ( { \\pmb x } )$ . This choice, as shown later, implies an interesting analogy to consistency regularization. The choice, also entails no distinction between the $M - 1$ predictive distributions. Therefore, we consider $\\pi _ { 2 } = \\cdot \\cdot \\cdot = \\pi _ { M } = { \\textstyle { \\frac { 1 - \\pi _ { 1 } } { M - 1 } } }$ in all our experiments. Finally, we scale the loss functions by a constant factor $Z = - ( 1 - \\pi _ { 1 } ) \\log ( 1 - \\pi _ { 1 } )$ . As we will see later, the role of this scaling is merely to strengthen the already existing and desirable behaviors of these losses as $\\pi _ { 1 }$ approaches zero and one. Formally, we have JS and GJS losses: ",
+        "bbox": [
+            173,
+            670,
+            826,
+            842
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0be372a169161093620cf0b8e4d186785ac302831f5de97419d4d9cdd003936b.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { J S } } ( y , f , x ) : = \\frac { D _ { \\mathrm { J S } _ { \\pi } } ( e ^ { ( y ) } , f ( \\tilde { x } ) ) } { Z } , \\quad \\mathcal { L } _ { \\mathrm { G J S } } ( y , f , x ) : = \\frac { D _ { \\mathrm { G J S } _ { \\pi } } ( e ^ { ( y ) } , f ( \\tilde { x } ^ { ( 2 ) } ) , \\dots , f ( \\tilde { x } ^ { ( M ) } ) ) } { Z }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            847,
+            795,
+            881
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "with $\\tilde { \\mathbf { x } } ^ { ( i ) } \\sim \\mathcal { A } ( \\mathbf { x } )$ . Next, we study the connection between JS and losses which are based on the robustness theory of Ghosh et al. [2]. ",
+        "bbox": [
+            174,
+            882,
+            825,
+            912
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 JS’s Connection to Robust Losses ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            446,
+            106
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Cross Entropy (CE) is the prevalent loss function for deep classifiers with remarkable successes. However, CE is prone to fitting noise [9]. On the other hand, Mean Absolute Error (MAE) is theoretically noise-robust [2]. Evidently, standard optimization algorithms struggle to minimize MAE, especially for more challenging datasets e.g. CIFAR-100 [3, 5]. Therefore, there have been several proposals that combine CE and MAE, such as Generalized CE (GCE) [3], Symmetric CE (SCE) [4], and Normalized CE (NCE+MAE) [5]. The rationale is for CE to help with the learning dynamics of MAE. Next, we show JS has CE and MAE as its asymptotes w.r.t. $\\pi _ { 1 }$ . ",
+        "bbox": [
+            173,
+            116,
+            826,
+            214
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. Let $\\pmb { p } \\in \\Delta ^ { K - 1 }$ , then ",
+        "bbox": [
+            176,
+            215,
+            416,
+            231
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f776f3575962fc1bf5bc1b90d1c4daca9ecbdd4e16545869ee4aa85f682d75e3.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { \\pi _ { 1 } \\to 0 } \\mathcal L _ { \\mathrm { J S } } ( e ^ { ( y ) } , p ) = H ( e ^ { ( y ) } , p ) , \\qquad \\operatorname* { l i m } _ { \\pi _ { 1 } \\to 1 } \\mathcal L _ { \\mathrm { J S } } ( e ^ { ( y ) } , p ) = \\frac { 1 } { 2 } \\| e ^ { ( y ) } - p \\| _ { 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            235,
+            233,
+            764,
+            265
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $H ( e ^ { ( y ) } , p )$ is the cross entropy of $e ^ { ( y ) }$ relative to $\\pmb { p }$ ",
+        "bbox": [
+            174,
+            268,
+            557,
+            285
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 depicts how JS interpolates between CE and MAE for $\\pi _ { 1 } \\in ( 0 , 1 )$ . The proposition reveals an interesting connection to state-of-the-art robust loss functions, however, there are important differences. SCE is not bounded (so it cannot be used in Theorem 1), and GCE is not symmetric, while JS and MAE are both symmetric and bounded. In Appendix B.3, we perform a dissection to better understand how these properties affect learning with noisy labels. GCE is most similar to JS and is compared further in Appendix B.4. ",
+        "bbox": [
+            173,
+            292,
+            826,
+            377
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A crucial difference to these other losses is that JS naturally extends to multiple predictive distributions (GJS). Next, we show how GJS generalizes JS by incorporating consistency regularization. ",
+        "bbox": [
+            173,
+            382,
+            825,
+            411
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 GJS’s Connection to Consistency Regularization ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            426,
+            550,
+            443
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Figure 1, it was shown how the consistency of the noisy labeled examples was reduced when the network overfitted to noise. The following proposition shows how GJS naturally encourages consistency in a single principled loss function. ",
+        "bbox": [
+            174,
+            452,
+            826,
+            494
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. Let $\\pmb { p } ^ { ( 2 ) } , \\ldots , \\pmb { p } ^ { ( M ) } \\in \\Delta ^ { K - 1 }$ with $M \\geq 3$ and $\\begin{array} { r } { \\bar { p } _ { > 1 } = \\frac { \\sum _ { j = 2 } ^ { M } \\pi _ { j } { p ^ { ( j ) } } } { 1 - \\pi _ { 1 } } } \\end{array}$ , then ",
+        "bbox": [
+            176,
+            498,
+            746,
+            521
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8122483a9d73dfb714dd7febb8018499c331ddf198d15fbb8e523edf1335e18c.jpg",
+        "text": "$$\n\\mathscr { L } _ { \\mathrm { G J S } } ( e ^ { ( y ) } , \\pmb { p } ^ { ( 2 ) } , \\dots , \\pmb { p } ^ { ( M ) } ) = \\mathscr { L } _ { \\mathrm { J S } _ { \\pi ^ { \\prime } } } ( e ^ { ( y ) } , \\bar { \\pmb { p } } _ { > 1 } ) + ( 1 - \\pi _ { 1 } ) \\mathscr { L } _ { \\mathrm { G J S } _ { \\pi ^ { \\prime \\prime } } } ( \\pmb { p } ^ { ( 2 ) } , \\dots , \\pmb { p } ^ { ( M ) } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            218,
+            525,
+            777,
+            545
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where π0 = [π1, 1 − π1]T and π00 = [π2,...,πM ]T . ",
+        "bbox": [
+            173,
+            549,
+            496,
+            570
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Importantly, Proposition 2 shows that GJS can be decomposed into two terms: 1) a JS term between the label and the mean prediction $\\bar { p } _ { > 1 }$ , and 2) a GJS term, but without the label. Figure 3 illustrates the effect of this decomposition. The first term, similarly to the standard JS loss, encourages the predictions’ mean to be closer to the label (Figure 3 middle). However, the second term encourages all predictions to be similar, that is, consistency regularization (Figure 3 right). ",
+        "bbox": [
+            173,
+            578,
+            825,
+            650
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.5 Noise Robustness ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            664,
+            333,
+            679
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, the robustness properties of JS and GJS are analyzed in terms of lower $( B _ { L } )$ and upper bounds $( B _ { U } )$ for the following theorem, which generalizes the results by Zhang et al. [3] to any bounded loss function, even with multiple predictive distributions. ",
+        "bbox": [
+            173,
+            689,
+            825,
+            731
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Undsatisfied for a loss symmetric noise with , then $\\begin{array} { r } { \\eta < \\frac { K - 1 } { K } } \\end{array}$ , $\\begin{array} { r } { \\{ B _ { L } \\leq \\sum _ { i = 1 } ^ { K } \\mathcal { L } ( e ^ { ( i ) } , { \\pmb x } , f ) \\leq B _ { U } , \\forall { \\pmb x } , f } \\end{array}$ $\\mathcal { L }$ ",
+        "bbox": [
+            174,
+            734,
+            823,
+            765
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07da65794670cae65942c7655d7f5ac64533939cf68a15919acad5083bf10359.jpg",
+        "text": "$$\n0 \\leq R _ { \\mathcal { L } } ^ { \\eta } ( f ^ { * } ) - R _ { \\mathcal { L } } ^ { \\eta } ( f _ { \\eta } ^ { * } ) \\leq \\eta \\frac { B _ { U } - B _ { L } } { K - 1 } , \\quad a n d \\quad - \\frac { \\eta ( B _ { U } - B _ { L } ) } { K - 1 - \\eta K } \\leq R _ { \\mathcal { L } } ( f ^ { * } ) - R _ { \\mathcal { L } } ( f _ { \\eta } ^ { * } ) \\leq 0 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            767,
+            805,
+            801
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A tighter bound $B _ { U } - B _ { L }$ , implies a smaller worst case risk difference of the optimal classifiers (robust when $B _ { U } = B _ { L }$ ). Importantly, while $\\mathcal { L } ( e ^ { ( i ) } , \\pmb { x } , f ) = \\mathcal { L } ( e ^ { ( i ) } , f ( \\pmb { x } ) )$ usually, this subtle distinction is useful for losses with multiple predictive distributions, see Equation 3. In Theorem 2 in Appendix C.3, we further prove the robustness of the proposed losses to asymmetric noise. ",
+        "bbox": [
+            174,
+            805,
+            825,
+            863
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For losses with multiple predictive distributions, the bounds in Theorem 1 and 2 must hold for any $_ { \\textbf { \\em x } }$ and $f$ , i.e., for any combination of $M - 1$ categorical distributions on $K$ classes. Proposition 3 provides such bounds for GJS. ",
+        "bbox": [
+            174,
+            868,
+            825,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3. GJS loss with $M \\leq K + 1$ satisfies $\\begin{array} { r } { B _ { L } \\le \\sum _ { k = 1 } ^ { K } \\mathcal { L } _ { \\mathrm { G J S } } ( e ^ { ( k ) } , \\pmb { p } ^ { ( 2 ) } , \\dots , \\pmb { p } ^ { ( M ) } ) \\le B _ { U } } \\end{array}$ for all $\\pmb { p } ^ { ( 2 ) } , \\ldots , \\pmb { p } ^ { ( M ) } \\in \\Delta ^ { K - 1 }$ , with the following bounds ",
+        "bbox": [
+            169,
+            88,
+            825,
+            121
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cac3a1436367f3e3058bfb257d5bc2f2951b2d508849c8b2bf2ec78c0b164509.jpg",
+        "text": "$$\nB _ { L } = \\sum _ { k = 1 } ^ { K } \\mathcal { L } _ { \\mathrm { G J S } } ( e ^ { ( k ) } , { \\pmb u } , \\ldots , { \\pmb u } ) , \\quad B _ { U } = \\sum _ { k = 1 } ^ { K } \\mathcal { L } _ { \\mathrm { G J S } } ( e ^ { ( k ) } , e ^ { ( 1 ) } , \\ldots , e ^ { ( M - 1 ) } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            245,
+            119,
+            751,
+            162
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { u } \\in \\Delta ^ { K - 1 }$ is the uniform distribution. ",
+        "bbox": [
+            178,
+            166,
+            473,
+            181
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note the bounds for the JS loss is a special case of Proposition 3 for $M = 2$ ",
+        "bbox": [
+            174,
+            183,
+            673,
+            196
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 1. ${ \\mathcal { L } } _ { \\mathrm { J S } }$ and ${ \\mathcal { L } } _ { \\mathrm { G J S } }$ are robust $B _ { L } = B _ { U , }$ ) in the limit of $\\pi _ { 1 }  1$ ",
+        "bbox": [
+            173,
+            199,
+            650,
+            215
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 1 is intuitive from Section 2.3 which showed that ${ \\mathcal { L } } _ { \\mathrm { J S } }$ is equivalent to the robust MAE in this limit and that the consistency term in Proposition 2 vanishes. ",
+        "bbox": [
+            174,
+            223,
+            823,
+            252
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Proposition 3, the lower bound $( B _ { L } )$ is the same for JS and GJS. However, the upper bound $( B _ { U } )$ increases for more distributions, which makes JS have a tighter bound than GJS in Theorem 1 and 2. In Proposition 4, we show that JS and GJS have the same bound for the risk difference, given an assumption based on Figure 1 that the optimal classifier on clean data $( f ^ { * } )$ is at least as consistent as the optimal classifier on noisy data $( f _ { \\eta } ^ { * } )$ . ",
+        "bbox": [
+            174,
+            257,
+            825,
+            329
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "${ \\mathcal { L } } _ { \\mathrm { J S } }$ ${ \\mathcal { L } } _ { \\mathrm { G J S } }$ ounds , where $^ { l }$ i f is $\\mathbb { E } _ { \\mathbf { x } } [ \\mathcal { L } _ { \\mathrm { G J S } _ { \\pi ^ { \\prime \\prime } } } ^ { f ^ { * } } ( \\pmb { p } ^ { ( 2 ) } , \\dots , \\pmb { p } ^ { ( M ) } ) ] \\le \\mathbb { E } _ { \\mathbf { x } } [ \\mathcal { L } _ { \\mathrm { G J S } _ { \\pi ^ { \\prime \\prime } } } ^ { f _ { \\eta } ^ { * } } ( \\pmb { p } ^ { ( 2 ) } , \\dots , \\pmb { p } ^ { ( M ) } ) ]$ $\\mathcal { L } _ { \\mathrm { G J S } _ { \\pi ^ { \\prime \\prime } } } ^ { f } ( \\pmb { p } ^ { ( 2 ) } , \\dots , \\pmb { p } ^ { ( M ) } )$ the consistency term from Proposition 2. ",
+        "bbox": [
+            174,
+            332,
+            825,
+            381
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Related Works ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            398,
+            330,
+            416
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Interleaved in the previous sections, we covered most-related works to us, i.e. the avenue of identification or construction of theoretically-motivated robust loss functions [2, 3, 4, 5]. These works, similar to this paper, follow the theoretical construction of Ghosh et al. [2]. Furthermore, Liu&Guo [10] use “peer prediction” to propose a new family of robust loss functions. Different to these works, here, we propose loss functions based on $D _ { \\mathrm { J S } }$ which holds various desirable properties of those prior works while exhibiting novel ties to consistency regularization; a recent important regularization technique. ",
+        "bbox": [
+            173,
+            426,
+            826,
+            511
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Next, we briefly cover other lines of work. A more thorough version can be found in Appendix D. ",
+        "bbox": [
+            176,
+            515,
+            815,
+            530
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A direction, that similar to us does not alter training, reweights a loss function by confusion matrix [11, 12, 13, 14, 15]. Assuming a class-conditional noise model, loss correction is theoretically motivated and perfectly orthogonal to noise-robust losses. ",
+        "bbox": [
+            174,
+            536,
+            825,
+            579
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Consistency regularization is a recent technique that imposes smoothness in the learnt function for semi-supervised learning [7] and recently for noisy data [16]. These works use different complex pipelines for such regularization. GJS encourages consistency in a simple way that exhibits other desirable properties for learning with noisy labels. Importantly, Jensen-Shannon-based consistency loss functions have been used to improve test-time robustness to image corruptions [6] and adversarial examples [17], which further verifies the general usefulness of GJS. In this work, we study such loss functions for a different goal: training-time label-noise robustness. In this context, our thorough analytical and empirical results are, to the best of our knowledge, novel. ",
+        "bbox": [
+            174,
+            584,
+            825,
+            696
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Recently, loss functions with information-theoretic motivations have been proposed [18, 19]. JS, with an apparent information-theoretic interpretation, has a strong connection to those. Especially, the latter is a close concurrent work studying JS and other divergences from the family of f-divergences [20]. However, in this work, we consider a generalization to more than two distributions and study the role of $\\pi _ { 1 }$ , which they treat as a constant $\\begin{array} { r } { \\check { \\boldsymbol { \\pi } } _ { 1 } = \\frac { 1 } { 2 } \\boldsymbol { \\cdot } } \\end{array}$ ). These differences lead to improved performance and novel theoretical results, e.g., Proposition 1 and 2. Lastly, another generalization of JS was recently presented by Nielsen [21], where the arithmetic mean is generalized to abstract means. ",
+        "bbox": [
+            174,
+            702,
+            825,
+            799
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            814,
+            312,
+            832
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This section, first, empirically investigates the effectiveness of the proposed losses for learning with noisy labels on synthetic (Section 4.1) and real-world noise (Section 4.2). This is followed by several experiments and ablation studies (Section 4.3) to shed light on the properties of JS and GJS through empirical substantiation of the theories and claims provided in Section 2. All these additional experiments are done on the more challenging CIFAR-100 dataset. ",
+        "bbox": [
+            174,
+            842,
+            823,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/251386a8af7e7753cb1efe1b1421813fdb685038184f84084bc48a0bbc526357.jpg",
+        "table_caption": [
+            "Table 1: Synthetic Noise Benchmark on CIFAR. We reimplement other noise-robust loss functions into the same learning setup and ResNet-34, including label smoothing (LS), Bootstrap (BS), Symmetric CE (SCE), Generalized CE (GCE), and Normalized CE $( \\mathrm { N C E + R C E } )$ ). We used same hyperparameter optimization budget and mechanism for all the prior works and ours. Mean test accuracy and standard deviation are reported from five runs and the statistically-significant top performers are boldfaced. The thorough analysis is evident from the higher performance of CE in our setup compared to prior works. GJS achieves state-of-the-art results for different noise rates, types, and datasets. Generally, GJS’s efficacy is more evident for the more challenging CIFAR-100 dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Method</td><td>No Noise</td><td colspan=\"4\">Symmetric Noise Rate</td><td colspan=\"2\">Asymmetric Noise Rate</td></tr><tr><td></td><td></td><td>0%</td><td>20%</td><td>40%</td><td>60%</td><td>80%</td><td>20%</td><td>40%</td></tr><tr><td></td><td>CE</td><td>95.77 ± 0.11</td><td>91.63 ±0.27</td><td>87.74 ±0.46</td><td>81.99 ± 0.56</td><td>66.51 ± 1.49</td><td>92.77 ± 0.24</td><td>87.12 ± 1.21</td></tr><tr><td></td><td>BS</td><td>94.58± 0.25</td><td>91.68 ±0.32</td><td>89.23 ±0.16</td><td>82.65 ± 0.57</td><td>16.97 ± 6.36</td><td>93.06±0.25</td><td>88.87 ± 1.06</td></tr><tr><td></td><td>LS</td><td>95.64 ± 0.12</td><td>93.51 ± 0.20</td><td>89.90 ±0.20</td><td>83.96±0.58</td><td>67.35 ± 2.71</td><td>92.94 ± 0.17</td><td>88.10 ±0.50</td></tr><tr><td>CIFAR-10</td><td>SCE</td><td>95.75 ± 0.16</td><td>94.29 ± 0.14</td><td>92.72 ±0.25</td><td>89.26± 0.37</td><td>80.68 ± 0.42</td><td>93.48± 0.31</td><td>84.98±0.76</td></tr><tr><td></td><td>GCE</td><td>95.75 ± 0.14</td><td>94.24 ± 0.18</td><td>92.82 ± 0.11</td><td>89.37 ±0.27</td><td>79.19 ± 2.04</td><td>92.83 ±0.36</td><td>87.00 ±0.99</td></tr><tr><td></td><td>NCE+RCE</td><td>95.36±0.09</td><td>94.27 ± 0.18</td><td>92.03 ±0.31</td><td>87.30 ± 0.35</td><td>77.89 ± 0.61</td><td>93.87 ± 0.03</td><td>86.83 ±0.84</td></tr><tr><td></td><td>JS</td><td>95.89 ±0.10</td><td>94.52 ± 0.21</td><td>93.01 ±0.22</td><td>89.64 ±0.15</td><td>76.06 ±0.85</td><td>92.18 ± 0.31</td><td>87.99 ± 0.55</td></tr><tr><td></td><td>GJS</td><td>95.91 ± 0.09</td><td>95.33 ± 0.18</td><td>93.57 ± 0.16</td><td>91.64 ± 0.22</td><td>79.11 ± 0.31</td><td>93.94±0.25</td><td>89.65± 0.37</td></tr><tr><td></td><td>CE</td><td>77.60 ± 0.17</td><td>65.74 ± 0.22</td><td>55.77 ± 0.83</td><td>44.42 ± 0.84</td><td>10.74 ± 4.08</td><td>66.85±0.32</td><td>49.45 ± 0.37</td></tr><tr><td></td><td>BS</td><td>77.65± 0.29</td><td>72.92 ± 0.50</td><td>68.52 ± 0.54</td><td>53.80 ± 1.76</td><td>13.83 ± 4.41</td><td>73.79 ± 0.43</td><td>64.67 ± 0.69</td></tr><tr><td></td><td>LS</td><td>78.60 ±0.04</td><td>74.88 ± 0.15</td><td>68.41 ± 0.20</td><td>54.58 ± 0.47</td><td>26.98 ± 1.07</td><td>73.17 ± 0.46</td><td>57.20±0.85</td></tr><tr><td>CIFAR-100</td><td>SCE</td><td>78.29±0.24</td><td>74.21 ± 0.37</td><td>68.23 ±0.29</td><td>59.28 ± 0.58</td><td>26.80 ± 1.11</td><td>70.86 ± 0.44</td><td>51.12 ± 0.37</td></tr><tr><td></td><td>GCE</td><td>77.65 ± 0.17</td><td>75.02 ± 0.24</td><td>71.54 ± 0.39</td><td>65.21 ± 0.16</td><td>49.68 ±0.84</td><td>72.13 ±0.39</td><td>51.50 ± 0.71</td></tr><tr><td></td><td>NCE+RCE</td><td>74.66 ± 0.21</td><td>72.39± 0.24</td><td>68.79 ±0.29</td><td>62.18 ± 0.35</td><td>31.63 ± 3.59</td><td>71.35 ± 0.16</td><td>57.80±0.52</td></tr><tr><td></td><td>JS</td><td>77.95± 0.39</td><td>75.41 ± 0.28</td><td>71.12 ± 0.30</td><td>64.36 ±0.34</td><td>45.05 ± 0.93</td><td>71.70 ± 0.36</td><td>49.36±0.25</td></tr><tr><td></td><td>GJS</td><td>79.27 ±0.29</td><td>78.05±0.25</td><td>75.71±0.25</td><td>70.15 ±0.30</td><td>44.49 ± 0.53</td><td>74.60 ± 0.47</td><td>63.70 ±0.22</td></tr></table>",
+        "bbox": [
+            174,
+            213,
+            823,
+            433
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Experimental Setup. We use ResNet 34 and 50 for experiments on CIFAR and WebVision datasets respectively and optimize them using SGD with momentum. The complete details of the training setup can be found in Appendix A. Most importantly, we take three main measures to ensure a fair and reliable comparison throughout the experiments: 1) we reimplement all the loss functions we compare with in a single shared learning setup, 2) we use the same hyperparameter optimization budget and mechanism for all the prior works and ours, and 3) we train and evaluate five networks for individual results, where in each run the synthetic noise, network initialization, and data-order are differently randomized. The thorough analysis is evident from the higher performance of CE in our setup compared to prior works. Where possible, we report mean and standard deviation and denote the statistically-significant top performers with student t-test. ",
+        "bbox": [
+            173,
+            455,
+            825,
+            593
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 Synthetic Noise Benchmarks: CIFAR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            609,
+            473,
+            625
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here, we evaluate the proposed loss functions on the CIFAR datasets with two types of synthetic noise: symmetric and asymmetric. For symmetric noise, the labels are, with probability $\\eta$ , resampled from a uniform distribution over all labels. For asymmetric noise, we follow the standard setup of Patrini et al. [22]. For CIFAR-10, the labels are modified, with probability $\\eta$ , as follows: truck automobile, $b i r d $ airplane, $c a t  d o g$ , and $d e e r \\to h o r s e$ . For CIFAR-100, labels are, with probability $\\eta$ , cycled to the next sub-class of the same “super-class”, e.g. the labels of super-class “vehicles $1 ^ { \\circ }$ are modified as follows: bicycle $ b u s  m o t o r c y c l e  p i c k u p t r u c k  t r a i n  b i c y c l e$ . ",
+        "bbox": [
+            173,
+            635,
+            826,
+            733
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We compare with other noise-robust loss functions such as label smoothing (LS) [23], Bootstrap (BS) [24], Symmetric Cross-Entropy (SCE) [4], Generalized Cross-Entropy (GCE) [3], and the $\\mathrm { N C E + R C E }$ loss of Ma et al. [5]. Here, we do not compare to methods that propose a full pipeline since, first, a conclusive comparison would require re-implementation and individual evaluation of several components and second, robust loss functions can be considered orthogonal to them. ",
+        "bbox": [
+            174,
+            738,
+            825,
+            808
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Results. Table 1 shows the results for symmetric and asymmetric noise on CIFAR-10 and CIFAR100. GJS performs similarly or better than other methods for different noise rates, noise types, and data sets. Generally, GJS’s efficacy is more evident for the more challenging CIFAR-100 dataset. For example, on $60 \\%$ uniform noise on CIFAR-100, the difference between GJS and the second best (GCE) is 4.94 percentage points, while our results on $80 \\%$ noise is lower than GCE. We attribute this to the high sensitivity of the results to the hyperparameter settings in such a high-noise rate which are also generally unrealistic (WebVision has $\\sim 2 0 \\%$ ). The performance of JS is consistently similar to the top performance of the prior works across different noise rates, types and datasets. In Section 4.3, we substantiate the importance of the consistency term, identified in Proposition 2, when going from JS to GJS that helps with the learning dynamics and reduce the susceptibility to noise. In Appendix B.1, we provide results for GJS on instance-dependent synthetic noise [25]. Next, we test the proposed losses on a naturally-noisy dataset to see their efficacy in a real-world scenario. ",
+        "bbox": [
+            174,
+            814,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/253523e2c57b69ed34e4f81acbc22a425e00fdee10e0183d01596d84e8d48cdb.jpg",
+        "table_caption": [
+            "Table 2: Real-world Noise Benchmark on WebVision. Mean test accuracy and standard deviation from five runs are reported for the validation sets of (mini) WebVision and ILSVRC12. GJS with two networks correspond to the mean prediction of two independently trained GJS networks with different seeds for data augmentation and weight initialization. Here, GJS uses $Z = 1$ . Results marked with $\\dagger$ are from Zheltonozhskii et al. [26]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Architecture</td><td rowspan=\"2\">Augmentation Networks</td><td rowspan=\"2\"></td><td colspan=\"2\">WebVision</td><td colspan=\"2\">ILSVRC12</td></tr><tr><td>Top1</td><td>Top5</td><td>Top1</td><td>Top 5</td></tr><tr><td>ELR+ [27]†</td><td>Inception-ResNet-V2</td><td>Mixup</td><td>2</td><td>77.78</td><td>91.68</td><td>70.29</td><td>89.76</td></tr><tr><td>DivideMix [16]t]</td><td>Inception-ResNet-V2</td><td>Mixup</td><td>2</td><td>77.32</td><td>91.64</td><td>75.20</td><td>90.84</td></tr><tr><td>DivideMix [16]t</td><td>ResNet-50</td><td>Mixup</td><td>2</td><td>76.32 ± 0.36</td><td>90.65 ± 0.16</td><td>74.42 ± 0.29</td><td>91.21±0.12</td></tr><tr><td>CE</td><td>ResNet-50</td><td>ColorJitter</td><td>1</td><td>70.69 ± 0.66</td><td>88.64 ± 0.17</td><td>67.32 ± 0.57</td><td>88.00 ±0.49</td></tr><tr><td>JS</td><td>ResNet-50</td><td>ColorJitter</td><td>1</td><td>74.56± 0.32</td><td>91.09 ±0.08</td><td>70.36 ± 0.12</td><td>90.60 ±0.09</td></tr><tr><td>GJS</td><td>ResNet-50</td><td>ColorJitter</td><td>1</td><td>77.99 ± 0.35</td><td>90.62 ±0.28</td><td>74.33 ± 0.46</td><td>90.33±0.20</td></tr><tr><td>GJS</td><td>ResNet-50</td><td>ColorJitter</td><td>2</td><td>79.28±0.24</td><td>91.22 ± 0.30</td><td>75.50 ± 0.17</td><td>91.27±0.26</td></tr></table>",
+        "bbox": [
+            173,
+            172,
+            830,
+            313
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            334,
+            825,
+            404
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 Real-World Noise Benchmark: WebVision ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            426,
+            508,
+            443
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "WebVision v1 is a large-scale image dataset collected by crawling Flickr and Google, which resulted in an estimated $20 \\%$ of noisy labels [28]. There are 2.4 million images of the same thousand classes as ILSVRC12. Here, we use a smaller version called mini WebVision [29] consisting of the first 50 classes of the Google subset. We compare CE, JS, and GJS on WebVision following the same rigorous procedure as for the synthetic noise. However, upon request by the reviewers, we also compare with the reported results of some state-of-the-art elaborate techniques. This comparison deviates from our otherwise systematic analysis. ",
+        "bbox": [
+            173,
+            455,
+            825,
+            553
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results. Table 2, as the common practice, reports the performances on the validation sets of WebVision and ILSVRC12 (first 50 classes). Both JS and GJS exhibit large margins with standard CE, especially for top-1 accuracy. Top-5 accuracy, due to its admissibility of wrong top predictions, can obscure the susceptibility to noise-fitting and thus indicates smaller but still significant improvements. ",
+        "bbox": [
+            174,
+            559,
+            825,
+            614
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The two state-of-the-art methods on this dataset were DivideMix [16] and $\\mathrm { E L R + }$ [27]. Compared to our setup, both these methods use a stronger network (Inception-ResNet-V2 vs ResNet-50), stronger augmentations (Mixup vs color jittering) and co-train two networks. Furthermore, $\\mathrm { E L R + }$ uses an exponential moving average of weights and DivideMix treats clean and noisy labeled examples differently after separating them using Gaussian mixture models. Despite these differences, GJS performs as good or better in terms of top-1 accuracy on WebVision and significantly outperforms $\\mathrm { E L R + }$ on ILSVRC12 (70.29 vs 74.33). The importance of these differences becomes apparent as 1) the top-1 accuracy for DivideMix degrades when using ResNet-50, and 2) the performance of GJS improves by adding one of their components, i.e. the use of two networks. We train an ensemble of two independent networks with the GJS loss and average their predictions (last row of Table 2). This simple extension, which requires no change in the training code, gives significant improvements. To the best of our knowledge, this is the highest reported top-1 accuracy on WebVision and ILSVRC12 when no pre-training is used. ",
+        "bbox": [
+            174,
+            621,
+            825,
+            800
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Appendix B.2, we show state-of-the-art results when using GJS on two other real-world noisy datasets: ANIMAL-10N [30] and Food-101N [31]. ",
+        "bbox": [
+            174,
+            808,
+            821,
+            835
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "So far, the experiments demonstrated the robustness of the proposed loss function (regarding Proposition 3) via the significant improvement of the final accuracy on noisy datasets. While this was central and informative, it is also important to investigate whether this improvement comes from the theoretical properties that were argued for JS and GJS. In what follows, we devise several such experiments, in an effort to substantiate the theoretical claims and conjectures. ",
+        "bbox": [
+            176,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/48861da0caca096c5aef4b7c6a2f45f415222a7bd0a52209f03c6370b8ac81e4.jpg",
+        "image_caption": [
+            "Figure 4: Effect of $\\pi _ { 1 }$ . Validation accuracy of JS and GJS during training with symmetric noise on CIFAR100. From Proposition 1, JS behaves like CE and MAE for low and high values of $\\pi _ { 1 }$ , respectively. The signs of noise-fitting for $\\pi _ { 1 } = 0 . 1$ on $60 \\%$ noise (b), and slow learning of $\\pi _ { 1 } = 0 . 9$ (a-b), show this in practice. The GJS loss does not exhibit overfitting for low values of $\\pi _ { 1 }$ and learns quickly for large values of $\\pi _ { 1 }$ (c-d). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            93,
+            480,
+            289
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/a67d759109b64075db34e39ff1fdc57bee0fbcaad6e7f81a5629a61ea0f843e6.jpg",
+        "image_caption": [
+            "Figure 5: Effect of M. Validation accuracy for increasing number of distributions $( M )$ and different symmetric noise rates on CIFAR-100 with $\\pi _ { 1 } = { \\frac { 1 } { 2 } } \\quad$ . For all noise rates, using three instead of two distributions results in a higher accuracy. Going beyond three distributions is only helpful for lower noise rates. For simplicity we use $M = 3$ (corresponding to two augmentations) for all of our experiments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            508,
+            108,
+            816,
+            289
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 Towards a Better Understanding of the Jensen-Shannon-based Loss Functions ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            426,
+            756,
+            441
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Here, we study the behavior of the losses for different distribution weights $\\pi _ { 1 }$ , number of distributions $M$ , and epochs. We also provide insights on why GJS performs better than JS. ",
+        "bbox": [
+            174,
+            448,
+            825,
+            476
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "How does $\\pi _ { 1 }$ control the trade-off of robustness and learnability? In Figure 4, we plot the validation accuracy during training for both JS and GJS at different values of $\\pi _ { 1 }$ and noise rates $\\eta$ From Proposition 1, we expect JS to behave as CE for low values of $\\pi _ { 1 }$ and as MAE for larger values of $\\pi _ { 1 }$ . Figure 4 (a-b) confirms this. Specifically, $\\pi _ { 1 } = 0 . 1$ learns quickly and performs well for low noise but overfits for $\\eta = 0 . 6$ (characteristic of non-robust CE), on the other hand, $\\pi _ { 1 } = 0 . 9$ learns slowly but is robust to high noise rates (characteristic of noise-robust MAE). ",
+        "bbox": [
+            174,
+            482,
+            825,
+            565
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Figure 4 (c-d), we observe three qualitative improvements of GJS over JS: 1) no signs of overfitting to noise for large noise rates with low values of $\\pi _ { 1 }$ , 2) better learning dynamics for large values of $\\pi _ { 1 }$ that otherwise learns slowly, and 3) converges to a higher validation accuracy. ",
+        "bbox": [
+            174,
+            571,
+            825,
+            613
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "How many distributions to use? Figure 5 depicts validation accuracy for varying number of distributions $M$ . For all noise rates, we observe a performance increase going from $M = 2$ to $M = 3$ However, the performance of $M > 3$ depends on the noise rate. For lower noise rates, having more than three distributions can improve the performance. For higher noise rates e.g. $6 0 \\%$ , having $M > 3$ degrades the performance. We hypothesise this is due to: 1) at high noise rates, there are only a few correctly labeled examples that can help guide the learning, and 2) going from $M = 2$ to $M = 3$ adds a consistency term, while $M > 3$ increases the importance of the consistency term in Proposition 2. Therefore, for a large enough M, the loss will find it easier to keep the consistency term low (keep predictions close to uniform as at the initialization), instead of generalizing based on the few clean examples. For simplicity, we have used $M = 3$ for all experiments with GJS. ",
+        "bbox": [
+            173,
+            619,
+            825,
+            758
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Is the improvements of GJS over JS due to mean prediction or consistency? Proposition 2 decomposed GJS into a JS term with a mean prediction $( { \\bar { p } } _ { > 1 } )$ and a consistency term operating on all distributions but the target. In Table 3, we compare the performance of JS and GJS to GJS without the consistency term, i.e., $\\mathcal { L } _ { \\mathrm { J S } _ { \\pi ^ { \\prime } } } ( e ^ { ( y ) } , \\bar { p } _ { > 1 } )$ . The results suggest that the improvement of GJS over JS can be attributed to the consistency term. ",
+        "bbox": [
+            174,
+            765,
+            825,
+            835
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 4 (a-b) showed that JS improves the learning dynamics of MAE by blending it with CE, controlled by $\\pi _ { 1 }$ . Similarly, we see here that the consistency term also improves the learning dynamics (underfitting and convergence speed) of MAE. Interestingly, Figure 4 (c-d), shows the higher values of $\\pi _ { 1 }$ (closer to MAE) work best for GJS, hinting that, the consistency term improves the learning dynamics of MAE so much so that the role of CE becomes less important. ",
+        "bbox": [
+            173,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4ee5bf73b1937cc97fd60ad8fa32c47d34bb822fa7e932a1bf71660a85cffc99.jpg",
+        "table_caption": [
+            "Table 3: Effect of Consistency. Validation accuracy for JS, GJS w/o the consistency term in Proposition 2, and GJS for $4 0 \\%$ noise on the CIFAR-100 dataset. Using the mean of two predictions in the JS loss does not improve performance. On the other hand, adding the consistency term significantly helps. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Accuracy</td></tr><tr><td>LJs(e(）,p（2））</td><td>71.0</td></tr><tr><td>LJsπ,(e(）,p&gt;1）</td><td>68.7</td></tr><tr><td>LGJs(e(y）,p(2）,p(3）)</td><td>74.3</td></tr></table>",
+        "bbox": [
+            225,
+            200,
+            434,
+            267
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/0c738eb8bd417ca80890d8f2086da3ee22f646ac3041bedf28d88816c70765fb.jpg",
+        "table_caption": [
+            "Table 4: Effect of GJS. Validation accuracy when using different loss functions for clean and noisy examples of the CIFAR-100 training set with $40 \\%$ symmetric noise. Noisy examples benefit significantly more from GJS than clean examples (74.1 vs 72.9). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Method</td><td colspan=\"3\">T1</td></tr><tr><td>Clean</td><td>Noisy</td><td>0.1</td><td>0.5</td><td>0.9</td></tr><tr><td>JS</td><td>JS</td><td>70.0</td><td>71.5</td><td>55.3</td></tr><tr><td>GJS</td><td>JS</td><td>72.6</td><td>72.9</td><td>70.2</td></tr><tr><td>JS</td><td>GJS</td><td>71.0</td><td>74.1</td><td>68.0</td></tr><tr><td>GJS</td><td>GJS</td><td>71.3</td><td>74.7</td><td>73.8</td></tr></table>",
+        "bbox": [
+            566,
+            179,
+            766,
+            275
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Is GJS mostly helping the clean or noisy examples? To better understand the improvements of GJS over JS, we perform an ablation with different losses for clean and noisy examples, see Table 4. We observe that using GJS instead of JS improves performance in all cases. Importantly, using GJS only for the noisy examples performs significantly better than only using it for the clean examples (74.1 vs 72.9). The best result is achieved when using GJS for both clean and noisy examples but still close to the noisy-only case (74.7 vs 74.1). ",
+        "bbox": [
+            173,
+            301,
+            825,
+            386
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "How is different choices of perturbations affecting GJS? In this work, we use stochastic augmentations for $\\mathcal { A }$ , see Appendix A.1 for details. Table 5 reports validation results on $40 \\%$ symmetric and asymmetric noise on CIFAR-100 for varying types of augmentation. We observe that all methods improve their performance with stronger augmentation and that GJS achieves the best results in all cases. Also, note that we use weak augmentation for all naturally-noisy datasets (WebVision, ANIMAL-10N, and Food-101N) and still get state-of-the-art results. ",
+        "bbox": [
+            173,
+            392,
+            825,
+            476
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "How fast is the convergence? We found that some baselines (especially the robust $\\mathrm { N C E + R C E }$ had slow convergence. Therefore, we used 400 epochs for all methods to make sure all had time to converge properly. Table 6 shows results on $40 \\%$ symmetric and asymmetric noise on CIFAR-100 when the number of epochs has been reduced by half. ",
+        "bbox": [
+            174,
+            482,
+            825,
+            537
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Is training with the proposed losses leading to more consistent networks? Our motivation for investigating losses based on Jensen-Shannon divergence was partly due to the observation in Figure 1 that consistency and accuracy correlate when learning with CE loss. In Figure 6, we compare CE, JS, and GJS losses in terms of validation accuracy and consistency during training on CIFAR-100 with $40 \\%$ symmetric noise. We find that the networks trained with JS and GJS losses are more consistent and has higher accuracy. In Appendix B.7, we report the consistency of the networks in Table 1. ",
+        "bbox": [
+            173,
+            544,
+            825,
+            627
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Summary of experiments in the appendix. Due to space limitations, we report several important experiments in the appendix. We evaluate the effectiveness of GJS on 1) instance-dependent synthetic noise (Section B.1), and 2) real-world noisy datasets ANIMAL-10N and Food-101N (Section B.2). We also investigate the importance of 1) losses being symmetric and bounded for learning with noisy labels (Section B.3), and 2) a clean vs noisy validation set for hyperparameter selection and the effect of a single set of parameters for all noise rates (Section B.5). ",
+        "bbox": [
+            173,
+            633,
+            825,
+            717
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/0087b0adde0be622b15865b47679014fae3cb5a0b5b0bef5f816619f25b4ccf5.jpg",
+        "table_caption": [
+            "Table 5: Effect of Augmentation Strategy. Validation accuracy for training w/o CutOut(-CO) or w/o RandAug(-RA) or w/o both(weak) on $40 \\%$ symmetric and asymmetric noise on CIFAR-100. All methods improves by stronger augmentations. GJS performs best for all types of augmentations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">Method</td><td colspan=\"4\">Symmetric</td><td colspan=\"4\">Asymmetric</td></tr><tr><td>Full</td><td>-CO</td><td>-RA</td><td>Weak Full</td><td></td><td>-C0</td><td>-RA</td><td>Weak</td></tr><tr><td>GCE</td><td>70.8</td><td>64.2</td><td>64.1</td><td>58.0</td><td>51.7</td><td>44.9</td><td>46.6</td><td>42.9</td></tr><tr><td>NCE+RCE</td><td>68.5</td><td>66.6</td><td>68.3</td><td>61.7</td><td>57.5</td><td>52.1</td><td>49.5</td><td>44.4</td></tr><tr><td>GJS</td><td>74.8</td><td>71.3</td><td>70.6</td><td>66.5</td><td>62.6</td><td>56.8</td><td>52.2</td><td>44.9</td></tr></table>",
+        "bbox": [
+            173,
+            832,
+            504,
+            916
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/c99688ea158fd74717d451dad33d7f7f23f53ff1cc2ed722fe8d05297f30694e.jpg",
+        "table_caption": [
+            "Table 6: Effect of Number of Epochs. Validation accuracy for training with 200 and 400 epochs for $40 \\%$ symmetric and asymmetric noise on CIFAR-100. GJS still outperforms the baselines and $\\mathrm { N C E + R C E }$ ’s performance is reduced heavily by the decrease in epochs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Symmetric</td><td colspan=\"2\">Asymmetric</td></tr><tr><td>200</td><td>400</td><td>200</td><td>400</td></tr><tr><td>GCE</td><td>70.3</td><td>70.8</td><td>39.1</td><td>51.7</td></tr><tr><td>NCE+RCE</td><td>60.0</td><td>68.5</td><td>35.0</td><td>57.5</td></tr><tr><td>GJS</td><td>72.9</td><td>74.8</td><td>43.2</td><td>62.6</td></tr></table>",
+        "bbox": [
+            524,
+            832,
+            820,
+            916
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/640df0ee49a168a2347bf5044f3ecb4712f30f38c30acf34b7bef3fac730e798.jpg",
+        "image_caption": [
+            "Figure 6: Evolution of a trained network’s consistency for the CE, JS, and GJS losses. We plot the evolution of the validation accuracy (a) and network’s consistency on clean (b) and noisy (c) examples of the training set of CIFAR-100 when learning with $40 \\%$ symmetric noise. All losses use the same learning rate and weight decay and both JS and GJS use $\\pi _ { 1 } = 0 . 5$ . The consistency of the learnt function and the accuracy closely correlate. The accuracy and consistency of JS and GJS improve during training, while both degrade when learning with CE loss. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            88,
+            812,
+            214
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Limitations & Future Directions ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            323,
+            475,
+            340
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We empirically showed that the consistency of the network around noisy data degrades as it fits noise and accordingly proposed a loss based on generalized Jensen-Shannon divergence (GJS). While we empirically verified the significant role of consistency regularization in robustness to noise, we only theoretically showed the robustness $B _ { L } = B _ { U , }$ ) of GJS at its limit $\\pi _ { 1 }  1 $ ) where the consistency term gradually vanishes. Therefore, the main limitation is the lack of a theoretical proof of the robustness of the consistency term in Proposition 2. This is, in general, an important but understudied area, also for the literature of self- or semi-supervised learning and thus is of utmost importance for future works. ",
+        "bbox": [
+            174,
+            351,
+            825,
+            462
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Secondly, we had an important observation that GJS with $M > 3$ might not perform well under high noise rates. While we have some initial conjectures, this phenomenon deserves a systematic analysis both empirically and theoretically. ",
+        "bbox": [
+            174,
+            468,
+            825,
+            511
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Finally, a minor practical limitation is the added computations for GJS forward passes, however this applies to training time only and in all our experiments, we only use one extra prediction ( $M = 3$ ). ",
+        "bbox": [
+            173,
+            516,
+            821,
+            545
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Final Remarks ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            564,
+            330,
+            582
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We first made two central observations that (i) robust loss functions have an underfitting issue and (ii) consistency of noise-fitting networks is significantly lower around noisy data points. Correspondingly, we proposed two loss functions, JS and GJS, based on Jensen-Shannon divergence that (i) interpolates between noise-robust MAE and fast-converging CE, and (ii) encourages consistency around training data points. This simple proposal led to state-of-the-art performance on both synthetic and real-world noise datasets even when compared to the more elaborate pipelines such as DivideMix or $\\mathrm { E L R + }$ . Furthermore, we discussed their robustness within the theoretical construction of Ghosh et al. [2]. By drawing further connections to other seminal loss functions such as CE, MAE, GCE, and consistency regularization, we uncovered other desirable or informative properties. We further empirically studied different aspects of the losses that corroborate various theoretical properties. ",
+        "bbox": [
+            173,
+            592,
+            826,
+            731
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Overall, we believe the paper provides informative theoretical and empirical evidence for the usefulness of two simple and novel JS divergence-based loss functions for learning under noisy data that achieve state-of-the-art results. At the same time, it opens interesting future directions. ",
+        "bbox": [
+            174,
+            737,
+            825,
+            779
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ethical Considerations. Considerable resources are needed to create labeled data sets due to the burden of manual labeling process. Thus, the creators of large annotated datasets are mostly limited to well-funded companies and academic institutions. In that sense, developing robust methods against label noise enables less affluent organizations or individuals to benefit from labeled datasets since imperfect or automatic labeling can be used instead. On the other hand, proliferation of such harvested datasets can increase privacy concerns arising from redistribution and malicious use. ",
+        "bbox": [
+            174,
+            785,
+            825,
+            868
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgement. This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. ",
+        "bbox": [
+            174,
+            883,
+            821,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Lucas Beyer, Olivier J Hénaff, Alexander Kolesnikov, Xiaohua Zhai, and Aäron van den Oord. Are we done with imagenet? arXiv preprint arXiv:2006.07159, 2020.   \n[2] Aritra Ghosh, Himanshu Kumar, and PS Sastry. Robust loss functions under label noise for deep neural networks. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, pages 1919–1925, 2017.   \n[3] Zhilu Zhang and Mert Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. In Advances in neural information processing systems, pages 8778–8788, 2018.   \n[4] Yisen Wang, Xingjun Ma, Zaiyi Chen, Yuan Luo, Jinfeng Yi, and James Bailey. Symmetric cross entropy for robust learning with noisy labels. In Proceedings of the IEEE International Conference on Computer Vision, pages 322–330, 2019.   \n[5] Xingjun Ma, Hanxun Huang, Yisen Wang, Simone Romano, Sarah Erfani, and James Bailey. Normalized loss functions for deep learning with noisy labels, 2020.   \n[6] Dan Hendrycks, Norman Mu, Ekin D. Cubuk, Barret Zoph, Justin Gilmer, and Balaji Lakshminarayanan. Augmix: A simple data processing method to improve robustness and uncertainty. In International Conference on Learning Representation, 2020.   \n[7] Avital Oliver, Augustus Odena, Colin Raffel, Ekin D Cubuk, and Ian J Goodfellow. Realistic evaluation of deep semi-supervised learning algorithms. arXiv preprint arXiv:1804.09170, 2018.   \n[8] Jianhua Lin. Divergence measures based on the shannon entropy. IEEE Transactions on Information theory, 37(1):145–151, 1991.   \n[9] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization, 2017.   \n[10] Yang Liu and Hongyi Guo. Peer loss functions: Learning from noisy labels without knowing noise rates. In Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 6226–6236. PMLR, 13–18 Jul 2020.   \n[11] Nagarajan Natarajan, Inderjit S Dhillon, Pradeep K Ravikumar, and Ambuj Tewari. Learning with noisy labels. In Advances in neural information processing systems, pages 1196–1204, 2013.   \n[12] Sainbayar Sukhbaatar, Joan Bruna, Manohar Paluri, Lubomir Bourdev, and Rob Fergus. Training convolutional networks with noisy labels. In Proceedings of the international conference on learning representation, 2015.   \n[13] Giorgio Patrini, Alessandro Rozza, Aditya Krishna Menon, Richard Nock, and Lizhen Qu. Making deep neural networks robust to label noise: A loss correction approach. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1944–1952, 2017.   \n[14] Bo Han, Jiangchao Yao, Gang Niu, Mingyuan Zhou, Ivor Tsang, Ya Zhang, and Masashi Sugiyama. Masking: A new perspective of noisy supervision. In Advances in Neural Information Processing Systems, pages 5836–5846, 2018.   \n[15] Xiaobo Xia, Tongliang Liu, Nannan Wang, Bo Han, Chen Gong, Gang Niu, and Masashi Sugiyama. Are anchor points really indispensable in label-noise learning? In Advances in Neural Information Processing Systems, pages 6838–6849, 2019.   \n[16] Junnan Li, Richard Socher, and Steven CH Hoi. Dividemix: Learning with noisy labels as semi-supervised learning. In International Conference on Learning Representation, 2020.   \n[17] Jihoon Tack, Sihyun Yu, Jongheon Jeong, Minseon Kim, Sung Ju Hwang, and Jinwoo Shin. Consistency regularization for adversarial robustness, 2021.   \n[18] Yilun Xu, Peng Cao, Yuqing Kong, and Yizhou Wang. L_dmi: A novel information-theoretic loss function for training deep nets robust to label noise. In Advances in Neural Information Processing Systems, pages 6225–6236, 2019.   \n[19] Jiaheng Wei and Yang Liu. When optimizing f-divergence is robust with label noise. In International Conference on Learning Representation, 2021.   \n[20] I. CSISZAR. Information-type measures of difference of probability distributions and indirect observation. Studia Scientiarum Mathematicarum Hungarica, 2:229–318, 1967.   \n[21] Frank Nielsen. On the jensen–shannon symmetrization of distances relying on abstract means. Entropy, 21(5), 2019.   \n[22] Giorgio Patrini, Alessandro Rozza, Aditya Menon, Richard Nock, and Lizhen Qu. Making deep neural networks robust to label noise: a loss correction approach, 2017.   \n[23] Michal Lukasik, Srinadh Bhojanapalli, Aditya Krishna Menon, and Sanjiv Kumar. Does label smoothing mitigate label noise? In International Conference on Machine Learning, 2020.   \n[24] Scott Reed, Honglak Lee, Dragomir Anguelov, Christian Szegedy, Dumitru Erhan, and Andrew Rabinovich. Training deep neural networks on noisy labels with bootstrapping. arXiv preprint arXiv:1412.6596, 2014.   \n[25] Yikai Zhang, Songzhu Zheng, Pengxiang Wu, Mayank Goswami, and Chao Chen. Learning with feature-dependent label noise: A progressive approach, 2021.   \n[26] Evgenii Zheltonozhskii, Chaim Baskin, Avi Mendelson, Alex M. Bronstein, and Or Litany. Contrast to divide: Self-supervised pre-training for learning with noisy labels, 2021.   \n[27] Sheng Liu, Jonathan Niles-Weed, Narges Razavian, and Carlos Fernandez-Granda. Earlylearning regularization prevents memorization of noisy labels, 2020.   \n[28] Wen Li, Limin Wang, Wei Li, Eirikur Agustsson, and Luc Van Gool. Webvision database: Visual learning and understanding from web data, 2017.   \n[29] Lu Jiang, Zhenyuan Zhou, Thomas Leung, Jia Li, and Fei-Fei Li. Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels. In ICML, 2018.   \n[30] Hwanjun Song, Minseok Kim, and Jae-Gil Lee. SELFIE: Refurbishing unclean samples for robust deep learning. In ICML, 2019.   \n[31] Kuang-Huei Lee, Xiaodong He, Lei Zhang, and Linjun Yang. Cleannet: Transfer learning for scalable image classifier training with label noise. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.   \n[32] Ekin D. Cubuk, Barret Zoph, Jonathon Shlens, and Quoc V. Le. Randaugment: Practical automated data augmentation with a reduced search space, 2019.   \n[33] Terrance DeVries and Graham W. Taylor. Improved regularization of convolutional neural networks with cutout, 2017.   \n[34] Junnan Li, Yongkang Wong, Qi Zhao, and Mohan Kankanhalli. Learning to learn from noisy labeled data, 2019.   \n[35] Duc Tam Nguyen, Chaithanya Kumar Mummadi, Thi Phuong Nhung Ngo, Thi Hoai Phuong Nguyen, Laura Beggel, and Thomas Brox. Self: Learning to filter noisy labels with selfensembling. In International Conference on Learning Representation, 2019.   \n[36] Curtis G Northcutt, Tailin Wu, and Isaac L Chuang. Learning with confident examples: Rank pruning for robust classification with noisy labels. arXiv preprint arXiv:1705.01936, 2017.   \n[37] Daiki Tanaka, Daiki Ikami, Toshihiko Yamasaki, and Kiyoharu Aizawa. Joint optimization framework for learning with noisy labels. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5552–5560, 2018.   \n[38] Arash Vahdat. Toward robustness against label noise in training deep discriminative neural networks. In Advances in Neural Information Processing Systems, pages 5596–5605, 2017.   \n[39] Ahmet Iscen, Giorgos Tolias, Yannis Avrithis, Ondrej Chum, and Cordelia Schmid. Graph convolutional networks for learning with few clean and many noisy labels. In Proceedings of the European Conference on Computer Vision, 2020.   \n[40] Paul Hongsuck Seo, Geeho Kim, and Bohyung Han. Combinatorial inference against label noise. In Advances in Neural Information Processing Systems, pages 1173–1183, 2019.   \n[41] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2818–2826, 2016.   \n[42] Takeru Miyato, Shin-ichi Maeda, Masanori Koyama, and Shin Ishii. Virtual adversarial training: a regularization method for supervised and semi-supervised learning. IEEE transactions on pattern analysis and machine intelligence, 41(8):1979–1993, 2018.   \n[43] David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin A Raffel. Mixmatch: A holistic approach to semi-supervised learning. In Advances in Neural Information Processing Systems, pages 5049–5059, 2019.   \n[44] Antti Tarvainen and Harri Valpola. Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. In Advances in neural information processing systems, pages 1195–1204, 2017. ",
+        "bbox": [
+            173,
+            73,
+            828,
+            916
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            54,
+            828,
+            916
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            90,
+            828,
+            415
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Checklist ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            439,
+            254,
+            455
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            465,
+            339,
+            479
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] See Section 6.   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            484,
+            825,
+            589
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            214,
+            593,
+            493,
+            608
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] See Section C in the Appendix. ",
+        "bbox": [
+            238,
+            612,
+            825,
+            656
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            214,
+            661,
+            393,
+            675
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See footnote on the first page.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section A in the Appendix.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See main experiments in Table 1 & 2.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section A. ",
+        "bbox": [
+            238,
+            679,
+            825,
+            811
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            223,
+            815,
+            825,
+            830
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes]   \n(b) Did you mention the license of the assets? [N/A]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] ",
+        "bbox": [
+            238,
+            834,
+            823,
+            912
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            236,
+            92,
+            823,
+            119
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            214,
+            123,
+            705,
+            138
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            142,
+            826,
+            232
+        ],
+        "page_idx": 13
+    }
+]

parse/train/Tt1s9Oi1kCS/Tt1s9Oi1kCS.md

@@ -0,0 +1,344 @@
+# CONTINUAL PROTOTYPE EVOLUTION: LEARNING ONLINE FROM NON-STATIONARY DATA STREAMS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Attaining prototypical features to represent class distributions is well established in representation learning. However, learning prototypes online from streams of data proves a challenging endeavor as they rapidly become outdated, caused by an ever-changing parameter space in the learning process. Additionally, continual learning does not assume the data stream to be stationary, typically resulting in catastrophic forgetting of previous knowledge. As a first, we introduce a system addressing both problems, where prototypes evolve continually in a shared latent space, enabling learning and prediction at any point in time. In contrast to the major body of work in continual learning, data streams are processed in an online fashion, without additional task-information, and an efficient memory scheme provides robustness to imbalanced data streams. Besides nearest neighbor based prediction, learning is facilitated by a novel objective function, encouraging cluster density about the class prototype and increased inter-class variance. Furthermore, the latent space quality is elevated by pseudo-prototypes in each batch, constituted by replay of exemplars from memory. We generalize the existing paradigms in continual learning to incorporate data incremental learning from data streams by formalizing a two-agent learner-evaluator framework, and obtain state-of-the-art performance by a significant margin on eight benchmarks, including three highly imbalanced data streams.
+
+# 1 INTRODUCTION
+
+The prevalence of data streams in contemporary applications urges systems to learn in a continual fashion. Autonomous vehicles, sensory robot data, and video streaming yield never-ending streams of data, with abrupt changes in the observed environment behind every vehicle turn, robot entering a new room, or camera cut to a subsequent scene. Alas, learning from streaming data is far from trivial due to these changes, as neural networks tend to forget the knowledge they previously acquired. The data stream presented to the network is not identically and independently distributed (iid), emanating a trade-off between neural stability to retain the current state of knowledge and neural plasticity to swiftly adopt the new knowledge (Grossberg, 1982). Finding the balance in this stability-plasticity dilemma addresses the catastrophic forgetting (French, 1999) induced by the non-iid intrinsics of the data stream, and is considered the main hurdle for continually learning systems.
+
+Although a lot of progress has been established in the literature, often strong assumptions apply, impeding applicability for real-world systems. The static training and testing paradigms prevail, whereas a true continual learner should enable both simultaneously and independently. Therefore, we propose the two-agent learner-evaluator framework to redefine perspective on existing paradigms in the field. Within this framework, we introduce data incremental learning, enabling completely task-free learning and evaluation.
+
+Furthermore, we introduce Continual Prototype Evolution (CoPE), a new online data incremental learner wherein prototypes perpetually represent the most salient features of the class population, shifting the catastrophic forgetting problem from the full network parameter space to the lowerdimensional latent space. As a first, our prototypes evolve continually with the data stream, enabling learning and evaluation at any point in time. Similar to representativeness heuristics in human cognition (Kahneman & Tversky, 1972), the class prototypes are the cornerstone for nearest neighbor classification. Additionally, the system is robust to highly imbalanced data streams by the combination of replay with a balancing memory population scheme. We find batch information in the latent space to have a significant advantage in the challenging non-stationary and online processing regime, which we incorporate in the novel pseudo-prototypical proxy loss.
+
+![](images/617ceb67615e434c3d5f24870ba9689683b9f285bccccf49b2d4dbcc366efa58.jpg)  
+Figure 1: Overview of the learner-evaluator framework, overcoming the static training and testing paradigms by explicitly modelling continual optimization and evaluation from data streams in the learner and evaluator agents. The framework generalizes to both continual learning and concept drift with resources transparently defined as the horizon $\mathcal { D }$ and operational memory $\mathcal { M }$ .
+
+# 2 THE LEARNER-EVALUATOR FRAMEWORK
+
+To date, the paradigms of task, class, and domain incremental learning (van de Ven & Tolias, 2018) dominate the continual learning literature. However, strong and differing assumptions often lead to confusion and overlap between implementations of these definitions. Furthermore, the concept of a static training and testing phase is still ubiquitous, whereas continual learning systems should enable both phases continually and independently. Therefore, we propose a generalizing framework which disentangles the continually learning system into two agents: the learner and the evaluator. Figure 1 presents an overview of the framework.
+
+The learning agent learns predicting function $f _ { \theta } : \mathcal { X }  \mathcal { V }$ parameterized by $\theta$ , mapping the input space $\mathcal { X }$ to the target output space $\mathcal { V }$ . The learner receives data samples $\left( \mathbf { x } _ { i } , \mathbf { y } _ { i } \right)$ from stream $S$ and has simultaneous access to the horizon $\mathcal { D }$ , i.e. the observable subset of stream $S$ which can be processed for multiple iterations. Data sample $i$ is constituted by input feature $\mathbf { x } _ { i } \in \mathcal { X }$ and corresponding (self-)supervision signal $\mathbf { y } _ { i }$ for which the output space for classification is defined as a discrete set of observed classes $\mathcal { V } _ { i }  \mathcal { V } _ { i - 1 } \cup \{ \mathbf { y } _ { i } \}$ . To manage memory usage and to enable multiple updates and stochasticity in the optimization process, updates for $\theta$ are typically performed based on a small-scale processing batch $B \subseteq { \mathcal { D } }$ . The data and size of the horizon $\mathcal { D }$ are determined by the specific setup or application, ranging from standard offline learning with $\mathcal { D } = S$ to online continual learning with $\mathcal { D } = B$ . Furthermore, the learner might need additional resources after observing data from $B \subseteq { \mathcal { D } }$ , such as stored samples or model copies, confined by the operational memory $\mathcal { M }$ .
+
+The evaluating agent acts independently from the learner by evaluating $f _ { \theta }$ with horizon $\mathcal { D } _ { e v a l }$ from the evaluation stream $S _ { e v a l }$ , with small-scale processing batches $B _ { e v a l } \ \subseteq \ D _ { e v a l }$ . This stream can contain yet unobserved concepts by the learner in $S$ to measure zero-shot performance. The framework provides leeway for the concept distributions in $S _ { e v a l }$ being either static or dynamically evolving, determining how performance of the learner is measured. On the one hand, static concept distributions can measure the degree to which the knowledge of learned concepts is preserved, as commonly used in continual learning. On the other hand, evolving concept distributions measure performance for the current distribution in horizon $\mathcal { D } _ { e v a l }$ only, where concepts might drift from their original representation, also known as concept drift (Schlimmer $\&$ Granger, 1986). Evaluation can occur asynchronously on-demand or periodically with periodicity $\rho$ determining the resolution of the evaluation samples.
+
+Task, class, and domain incremental learning are based on the composition in the learner for the observable stream subset in horizon $\mathcal { D } _ { t }$ , which is incrementally replaced by a new subset of data for the new task, set of classes, or domain, with $t$ the identifier of the present data subset. Task incremental learning assumes both learner and evaluator to get data $\left( \mathbf { x } _ { i } , \mathbf { y } _ { i } , t _ { i } \right)$ with $t _ { i + 1 } \geq t _ { i }$ and the horizon spanning all data of a given task with ${ \mathcal { D } } _ { t } = \{ ( \mathbf { x } _ { i } , \mathbf { y } _ { i } , t _ { i } ) \in S \mid t _ { i } = t \}$ (De Lange et al., 2019; van de Ven & Tolias, 2019). Having explicit access to $t _ { i }$ confines prediction to an isolated output space. Similarly, in class incremental learning the learner implicitly requires $t _ { i }$ to identify the transitions of $\mathcal { D }$ , when observing new batches of classes (Rebuffi et al., 2017; Castro et al., 2018; Shmelkov et al., 2017; Wu et al., 2018). However, the evaluator considers the entire output space without the need for identifier $t$ . Domain incremental learning holds the same assumptions as class incremental learning, with concepts drifting from one domain to the other for a typically fixed output space, exemplified by the widely used permuted-MNIST setup (Goodfellow et al., 2013).
+
+Data incremental learning is a more general paradigm we introduce to facilitate learning from any data stream, with no assumption but to observe data incrementally. In contrast to existing paradigms, when the learner observes horizon $\mathcal { D }$ of data stream $S$ , data incremental learning does not disclose an identifier $t$ . Consequently, there is no explicit indication to which subset of the stream is being observed in the horizon $\mathcal { D }$ . Therefore, the learner either processes observed data directly in an online fashion with processing batch $B = \mathcal { D }$ , or infers an implicit identifier $t$ from statistics in stream $S$ Similar to class and domain incremental learning, the evaluator operates without $t$ on the full output space. This paradigm endows continually learning systems with increased practical use, as real-world streaming applications often lack supervision signal $t$ . Moreover, even if $t$ is provided, this would introduce a bias in the fixed choice of the supervisor, rather than dynamically determined based on the needs of the system.
+
+# 3 PRIOR WORK
+
+Continually learning systems are able to learn with limited resources from data streams prone to severe distribution shifts. The main body of works presumes the presence of tasks, which divide the data streams into large discrete batches, and are indicated to the learner with a task identifier (Kirkpatrick et al., 2017; Li & Hoiem, 2017; Zenke et al., 2017; Aljundi et al., 2018; De Lange et al., 2020). Replay methods retain representative data for observed data distributions, currently unavailable in the learner’s horizon $\mathcal { D }$ . The replay data is either obtained directly from operational memory $\mathcal { M }$ with stored samples (Rebuffi et al., 2017; Lopez-Paz & Ranzato, 2017) or generated using generative models (Shin et al., 2017; Kamra et al., 2017; Seff et al., 2017; Wu et al., 2018). GEM (Lopez-Paz & Ranzato, 2017) uses replay in a constraint optimization perspective to project gradients towards a local joint task optimum. iCaRL (Rebuffi et al., 2017) employs exemplars to distill knowledge (Hinton et al., 2015) to the learner from a previous model version, with new class exemplars stored in a queue to optimally represent the class mean in feature space. The prototypes are then used for nearest neighbor prediction by the evaluator, in the same vein as concurrent work to ours (Han et al., 2020). Nonetheless, all three works strongly rely on task identifier $t$ for the learner, mostly unavailable for real-world data streams. Moreover, in both prototypical approaches the prototypes remain static between the given task transitions and become outdated. Consequently, before using the evaluator they have to exhaustively recalculate the prototypes based on all exemplars in memory. In contrast, our prototypes evolve in an online fashion with the data stream and remain representative for the continual learner and evaluator at all times.
+
+Recent works focus on online data incremental learning (Section 2) in which the learner operates completely task-free. Reservoir (Vitter, 1985) is a replay baseline with strong potential to outperform continual learning methods (Chaudhry et al., 2019). Samples are stored in memory $\mathcal { M }$ with probability $M / n$ , with $n$ the number of observed samples and buffer size $M$ . MIR (Aljundi et al., 2019a) extends Reservoir sampling with a loss-based retrieval strategy, with the cost of additional forward passes and a model copy to attain the losses for a subset of samples. The Reservoir buffer population approximately follows the data stream distribution, severely deteriorating the performance of underrepresented tasks in imbalanced data streams, as shown in Section 6.2. An alternative memory population scheme is used in GSS (Aljundi et al., 2019b) by extending the GEM constraint optimization perspective to an instance-based level. GSS adds samples to the buffer based on their gradients, whereas GEM requires the number of tasks and the task transitions to divide memory equally over all tasks a priori. In contrast, iCaRL’s memory population is incrementally subdivided over all classes after learning a task, by iteratively adding observed samples from $\mathcal { D }$ to optimally approximate the class mean in feature space. As this is computationally expensive, concurrent works to ours explore other balancing schemes (Kim et al., 2020; Chrysakis & Moens, 2020), where we propose a simple but effective class-based Reservoir scheme with uniform retrieval.
+
+![](images/6dc144b5152753f123b112cf2fc24956c7159796749fa6557baddf51f661dbaa.jpg)  
+Figure 2: Main setup. The learner updates network $f _ { \theta }$ and prototypes $\mathbf { p } ^ { y }$ $, \forall y \in \mathcal { V }$ continually. The PPP-loss encourages inter-class variance (red arrows) and reduces intra-class variance (green arrows).
+
+Another branch of parameter isolation methods (De Lange et al., 2019) allocates parameters to subsets of the data. Several task incremental works assign parameters based on the task identifier (Mallya & Lazebnik, 2018; Serra et al., 2018). A new line of work instead focuses on task-free model expansion. CURL (Rao et al., 2019) enables task-free and unsupervised adaptation using a multi-component variational auto-encoder, with generative replay from a model copy avoiding forgetting in the current model. CN-DPM (Lee et al., 2020) allocates data subsets to expert networks following a Dirichlet process mixture. In contrast to these capacity expansion based methods, CoPE evades unbound allocation of resources, as the memory and network capacity are fixed with the replay memory dynamically subdivided over categories occurring in the data stream. Note that new categories require an additional prototype, but these are only $d$ -dimensional and therefore insignificant in size, and the set of categories is typically limited as well.
+
+Besides the focus on continual learning in this work, our learner-evaluator framework generalizes to concept drift as well (Schlimmer & Granger, 1986), for which we refer to an overview in (Tsymbal, 2004; Gama et al., 2014). Further, in deep embedding learning most commonly pairs (Hadsell et al., 2006) and triplets (Harwood et al., 2017) of samples are considered in contrastive losses, whereas other works use batch information in lifted structure embeddings (Oh Song et al., 2016), or instancewise softmax embeddings (Ye et al., 2019). These approaches fully depend on the batch size, whereas our pseudo-prototypical proxy loss aggregates both decoupled prototypes and the additional batch pseudo-prototypes to defy class interference in the latent space. Learning prototypical representations also shows promising results in few-shot learning (Snell et al., 2017).
+
+# 4 CONTINUAL PROTOTYPE EVOLUTION
+
+The online data incremental learning setup of the learner is described in Figure 2. Embedding network $f _ { \theta }$ maps processing batch $B$ , composed of samples in horizon $\mathcal { D }$ from the non-iid data stream $S$ and operational memory $\mathcal { M }$ , to low-dimensional $\mathbb { R } ^ { d }$ latent space, followed by a nearest neighbor classifier. We enforce $| | f _ { \theta } ( \mathbf { \bar { x } } _ { i } ) | | = 1$ with an L2 normalization layer. $\mathcal { M }$ is subdivided in a replay memory $\mathcal { M } _ { r }$ and prototypical memory $\mathcal { M } _ { p }$ . CoPE comprises three main components: continually evolving representations, balanced replay and the pseudo-prototypical proxy (PPP) loss. In the following, we discuss these components and formalize the optimal choice of prototype, with $\mathbf { f } _ { i } ^ { c }$ denoting latent space projection $f _ { \theta } ( \mathbf { x } _ { i } ^ { c } )$ for an instance $\mathbf { x } _ { i }$ of class $c$ . For the full algorithm, we refer to Appendix A.
+
+# 4.1 EVOLVING REPRESENTATIONS
+
+Each observed class $c \in \mathcal { V }$ is represented by a slowly progressing prototype $\mathbf { p ^ { c } }$ in operational memory $\mathcal { M } _ { p }$ . The nearest neighbor classifier finds the most similar prototype for the given query $\mathbf { x } _ { i }$ , predicting $c ^ { * } = \arg \operatorname* { m a x } _ { c \in \mathcal { Y } } \mathbf { f _ { i } ^ { T } } \mathbf { p ^ { c } }$ . Similar to (Mensink et al., 2013; Rebuffi et al., 2017), the class-prototype approximates the center of mass in the latent space, which we formally justify in Section 4.4. The main crux with storing representations is to prevent them from becoming obsolete as the embedding network evolves. Additionally, this is further complicated by the shifting data distributions in the non-stationary regime, incurring catastrophic forgetting. Experience replay from a buffer $\mathcal { M } _ { r }$ is a well known approach to address this forgetting. Nonetheless, in our setup the replayed exemplars gain additional information about the current state of the embedding space, enabling rehearsal to rectify approximation $\mathbf { p } ^ { c }$ to the true center of mass. Concretely, the sampled batch
+
+$B _ { n }$ equals the horizon $\mathcal { D }$ from data stream $S$ and joins batch $B _ { \mathcal { M } }$ of equal size from memory $\mathcal { M } _ { r }$ , constituting $B$ as $B _ { n } \cup B _ { { \cal M } }$ . However, updating the prototypes by fully relying on features extracted from $B$ incurs an unstable optimization process as the representative prototypes depend on stochastic sampling of the class distributions. Therefore, we design the prototypes to evolve continually with a high momentum based update for each observed batch, aiming to stabilize the impetuous changes in the data stream:
+
+$$
+\mathbf { p } ^ { c }  \alpha \mathbf { p } ^ { c } + ( 1 - \alpha ) \bar { \mathbf { p } } ^ { c } , \mathrm { s . t . } \bar { \mathbf { p } } ^ { c } = \frac { 1 } { | B ^ { c } | } \sum _ { \mathbf { x } ^ { c } \in B ^ { c } } f _ { \boldsymbol \theta } ( \mathbf { x } ^ { c } ) ,
+$$
+
+with momentum parameter $\alpha \in [ 0 , 1 ]$ , the batch subset $B ^ { c } = \{ ( \mathbf { x } _ { i } , y _ { i } = c ) \in B \}$ of class $c$ , and $\bar { \mathbf { p } } ^ { c }$ the corresponding center of mass in latent space for the current batch. Due to triangle inequality $\mathbf { p } ^ { c }$ is no longer unit length and requires to be L2-normalized after the update in Eq. 1. We empirically validate the effectiveness of high momentum with $\alpha \approx 1$ in the ablation study in Appendix D.
+
+# 4.2 BALANCED REPLAY
+
+Similar to Rebuffi et al. (2017); Chrysakis & Moens (2020), the total buffer size $M$ is equally divided over the number of observed classes $| \mathcal { V } |$ in a dynamic fashion. This scheme ensures consistent buffer capacity for all classes, making memory allocation independent of the data stream characteristics. As $S$ is typically highly imbalanced in real-world scenarios, this memory scheme prevents classes to be eradicated from the buffer and assumes equal importance to represent each class at all times. Consequently, random retrieval from the buffer resembles sampling an iid replay batch. Furthermore, each class-specific replay memory $\mathcal { M } _ { r } ^ { c }$ can simply capture a random subset of its parent class distribution to approximate its center of mass. This avoids computationally expensive herding techniques as in iCaRL (Rebuffi et al., 2017), which would require recalculation of the feature means on each change of the memory size or network parameters.
+
+# 4.3 PSEUDO-PROTOTYPICAL PROXY LOSS
+
+The learner optimizes $f _ { \theta }$ to project an instance $\mathbf { f } _ { i } ^ { c } \in \mathbb { R } ^ { d }$ of class $c$ close to its corresponding prototype $\mathbf { p } ^ { c }$ in the latent space. As the prototype acts as a surrogate for the class mean in latent space, the cluster population has a common reference point to reduce intra-class variance, and enforce inter-class variance by remaining distant from the other class prototypes. Additionally, due to the embedding architecture we can use intrinsic information of the batch samples in the latent space. Therefore, we exploit the supervision signal $\mathbf { y } _ { i }$ in a sample $( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \in B$ not only to indicate which class $\mathbf { x } _ { i }$ belongs to, but also to make the distinction between positive and negative pairs in $B$ . Consequently, we can define one-against-all subsets for an instance of class $c$ , with positives from the same class in $B ^ { c } = \{ ( \mathbf { x } _ { i } , y _ { i } \stackrel { - } { = } c ) \in B \}$ and negatives in $B ^ { k }$ . Starting from these sets, the prototypical attractor and repellor sets for an instance $\mathbf { x } _ { i } ^ { c }$ are constituted with the class prototype $\mathbf { p } ^ { c }$ and the other instances in $B$ . First, the other instances of class $c$ act as pseudo-prototypes $\hat { \mathbf { p } } ^ { c }$ in attractor set $\mathbb { P } _ { i } ^ { c } = \{ \mathbf { p } ^ { c } \} \cup \{ \hat { \mathbf { p } } _ { j } ^ { c } = f _ { \theta } ( \mathbf { x } _ { j } ^ { c } ) \mid \forall \mathbf { x } _ { j } ^ { c } \in B ^ { c } , i \neq j \} _ { }$ . Second, the samples of other classes $\mathbf { x } _ { j } ^ { k } \in B ^ { k }$ should instead avoid both $\mathbf { x } _ { i } ^ { c }$ in latent space and the class representative $\mathbf { p } ^ { c }$ , defined by repellor set $\mathbb { U } _ { i } ^ { c } = \{ \mathbf { p } ^ { c } , \hat { \mathbf { p } } _ { i } ^ { c } = f _ { \theta } \big ( \mathbf { x } _ { i } ^ { c } \big ) \}$ . The attractor set for $\mathbf { x } _ { i } ^ { c }$ facilitates a decrease in intra-class variance with $\mathbf { p } ^ { c }$ safeguarding the absence of positive batch pairs with $1 \leq | \mathbb { P } _ { i } ^ { c } | \leq | B ^ { c } |$ , whereas the repellor exploits $\mathbf { x } _ { i } ^ { c }$ and corresponding prototype as a reference point to increase inter-class variance. To incorporate the attractor and repellor sets, we formulate a binary classification problem similar to Ye et al. (2019), with the joint probability that instance $\mathbf { x } _ { i } ^ { c }$ is predicted as class $c$ and instances $\mathbf { x } _ { j } ^ { k } \in B ^ { k }$ not being predicted as class $c$
+
+$$
+P _ { i } = P ( c | \mathbf { x } _ { i } ^ { c } ) \prod _ { \mathbf { x } _ { j } ^ { k } } ( 1 - P _ { i } ( c | \mathbf { x } _ { j } ^ { k } ) )
+$$
+
+with the assumption of independence between $\mathbf { x } _ { i } ^ { c }$ and $\mathbf { x } _ { j } ^ { k }$ being recognized as $c$ . We define the expected posterior probabilities for the attractor and repellor sets of instance $\mathbf { x } _ { i } ^ { c }$ respectively as
+
+$$
+\begin{array} { r } { P ( c | \mathbf { x } _ { i } ^ { c } ) = \underset { \tilde { \mathbf { p } } ^ { c } \in \mathbb { P } _ { i } ^ { c } } { \mathbb { E } } \left[ P ( c | \mathbf { f } _ { i } ^ { c } , \tilde { \mathbf { p } } ^ { c } ) \right] , \quad P _ { i } ( c | \mathbf { x } _ { j } ^ { k } ) = \underset { \tilde { \mathbf { p } } ^ { c } \in \mathbb { U } _ { i } ^ { c } } { \mathbb { E } } \left[ P ( c | \mathbf { f } _ { j } ^ { k } , \tilde { \mathbf { p } } ^ { c } ) \right] , } \end{array}
+$$
+
+with $\tilde { \mathbf { p } } ^ { c }$ a proxy for the latent mean of class $c$ in
+
+$$
+P ( c | \mathbf { f } , \tilde { \mathbf { p } } ^ { c } ) = \frac { \exp ( \mathbf { f } ^ { T } \tilde { \mathbf { p } } ^ { c } / \tau ) } { \exp ( \mathbf { f } ^ { T } \tilde { \mathbf { p } } ^ { c } / \tau ) + \sum _ { k \neq c } \exp ( \mathbf { f } ^ { T } \mathbf { p } ^ { k } / \tau ) } ,
+$$
+
+where temperature $\tau$ controls the concentration level of the distribution (Hinton et al., 2015), assuming a cosine similarity metric $\mathbf { f } _ { i } ^ { T } \mathbf { f } _ { j }$ with vectors normalized to unit length. We reformulate the objective in Eq.(2) as loss function $\mathcal { L }$ by negative log-likelihood and summation over all the instances in $B$ which approximates the true joint probability with assumed independent pairs in the batch:
+
+$$
+\mathcal { L } = - \frac { 1 } { | B | } \left[ \sum _ { i } \log P ( c | \mathbf { x } _ { i } ^ { c } ) + \sum _ { i } \sum _ { \mathbf { x } _ { j } ^ { k } } \log ( 1 - P _ { i } ( c | \mathbf { x } _ { j } ^ { k } ) ) \right] .
+$$
+
+# 4.4 OPTIMAL PROTOTYPES
+
+We update prototypes to approximate the mean of the parent distribution in Eq.(1). This assumption is optimal for Bregman divergences for which the cluster mean is shown to have minimal distance to its population (Banerjee et al., 2005). This Bregman divergence is defined for a differentiable, strictly convex function $\varphi$ as
+
+$$
+d _ { \varphi } ( \mathbf { f } _ { i } , \mathbf { f } _ { j } ) = \varphi ( \mathbf { f } _ { i } ) - \varphi ( \mathbf { f } _ { j } ) - ( \mathbf { f } _ { i } - \mathbf { f } _ { j } ) ^ { T } \nabla \varphi ( \mathbf { f } _ { j } ) ,
+$$
+
+for which the squared Euclidean distance with $\varphi ( \mathbf { f } ) = | | \mathbf { f } | | ^ { 2 }$ is a canonical example. The squared Euclidean distance is proportional to the cosine distance with vectors normalized to unit length: $\begin{array} { r } { \frac { 1 } { 2 } | | { \bf f } _ { i } - { \bf f } _ { j } | | ^ { 2 } = 1 - \cos \angle ( { \bf \hat { f } } _ { i } , { \bf f } _ { j } ) } \end{array}$ . As the PPP-loss in Eq.(4) requires a similarity measure instead of a distance measure, we employ the complementary normalized cosine similarity $\cos \angle ( { \bf f } _ { i } , { \bf f } _ { j } ) = { \bf f } _ { i } ^ { T } { \bf f } _ { j }$ with $| | \mathbf { f } _ { i } | | = | | \mathbf { f } _ { j } | | = 1$ . Besides the desirable cluster-mean property of its complement, this metric is also efficient for calculating the full batch similarity matrix using matrix multiplication libraries.
+
+# 5 EXPERIMENTS
+
+This work examines five balanced data streams and 15 highly imbalanced variants based on SplitMNIST, Split-CIFAR10 and Split-CIFAR100, from which two low-capacity balanced setups are discussed in Appendix E. The learner is presented a data stream $S$ , constituted by a sequence of tasks, each delineated by a subset of classes from the original dataset. Although the learner in CoPE is completely ignorant to the notion of task, this setup enables comparing to methods requiring task boundaries such as GEM and iCaRL. The evaluator uses a held-out dataset of static concepts in $S _ { e v a l }$ , evaluating with the subset of seen concepts $\mathcal { V }$ in $\mathcal { D } _ { e v a l }$ using the accuracy metric. The CoPE learner processes data online with $B _ { n } = \mathcal { D }$ in the data incremental setup, allowing per-task processing of 1 epoch for methods requiring task boundaries with $B \subset \mathcal { D }$ . We use vanilla stochastic gradient descent with a limited processing batch size $| B _ { n } |$ of 10 as in in (Lopez-Paz & Ranzato, 2017; Aljundi et al., 2019b; Lee et al., 2020). All results are averaged over 5 different network initializations.1
+
+Balanced data streams contain a similar amount of data per task. We consider three benchmarks. First, Split-MNIST constitutes the MNIST (LeCun et al., 1998) handwritten digit recognition dataset with 60k training samples, split into 5 tasks according to pairs of incrementing digits. Second, Split-CIFAR10 considers the CIFAR10 (Krizhevsky et al., 2009) dataset, subdivided into 5 tasks with 2 labels each, where each task entails 10k training samples. Third, Split-CIFAR100 is a variant of the CIFAR dataset with 100 different classes. The 50k training samples are subdivided in 20 tasks of $2 . 5 \mathrm { k }$ samples as in (Lopez-Paz & Ranzato, 2017; Lee et al., 2020). For all datasets the evaluator considers the entire original test subset for $S _ { e v a l }$ .
+
+Imbalanced data streams introduce a more realistic scenario without equality assumptions on the task durations in $S$ and address the literature mostly balancing the data streams artificially. Besides the imbalanced Split-MNIST setup (Aljundi et al., 2019b), we introduce two novel and more challenging benchmarks based on Split-CIFAR10 and Split-CIFAR100, where data stream $S$ comprises significantly more data in task $T _ { i }$ , denoted by $S ( T _ { i } )$ . Split-MNIST and Split-CIFAR10 have respectively $2 \mathrm { k }$ and 4k samples in $T _ { i }$ , whereas tasks $T _ { j }$ for $j \neq i$ contain factor 10 less data for five variants $S ( T _ { i } )$ , $\forall i \in \{ 1 , . . . , 5 \}$ . Split-CIFAR100 defines $T _ { i }$ with $2 . 5 \mathrm { k }$ samples and 1k for the remaining tasks, with variants $i \in \{ 1 , 5 , . . . , 2 0 \}$ .
+
+Architectures. MNIST setups use an MLP with 2 hidden layers of 400 units with $2 \mathrm { k }$ memories for the balanced setup as in (Hsu et al., 2018; Lee et al., 2020; van de Ven & Tolias, 2019), and 100 units with $| \mathcal { M } | = 0 . 3 \mathrm { k }$ for the imbalanced setup as in (Aljundi et al., 2019b). CIFAR setups use a slim version of Resnet18 (He et al., 2016) with a 1k memory size for CIFAR10 (Aljundi et al., 2019b; Lee et al., 2020), and $5 \mathrm { k }$ for CIFAR100 (Lopez-Paz & Ranzato, 2017).
+
+Methods compared to CoPE entail 11 baselines, with details on prior work discussed in Section 3. The upper reference point for performance when relaxing the challenging non-iid feature in continual learning is set by iid-online & iid-offline. The learner shuffles the full data stream $S$ to ensure the iid property, for which iid-online trains a single epoch and iid-offline multiple epochs. In contrast, the finetune learner considers non-iid data stream $S$ sequentially, but optimizes solely for the new batch which typically results in worst-case catastrophic forgetting. CoPE-CE is a reference point for the merits of a prototypical approach by solely using the CoPE memory and sampling scheme, but with a typical cross-entropy loss and softmax classifier. GEM and iCaRL are standard replay methods considered in a class incremental setup, with the learner requiring task boundaries. For online data incremental learning, we consider the reservoir, MIR and greedy GSS replay baselines, with CURL and CN-DPM instead relying on model expansion.
+
+Table 1: The three balanced data stream accuracies $( \% )$ with standard deviation over 5 initializations. Expansion-based methods CURL and DN-CPM report results from their original work.   
+
+<table><tr><td></td><td>Split-MNIST</td><td>Split-CIFAR10</td><td>Split-CIFAR100</td></tr><tr><td>iid-offline</td><td>98.44 ± 0.02</td><td>83.02 ±0.60</td><td>50.28±0.66</td></tr><tr><td>iid-online</td><td>96.57 ± 0.14</td><td>62.31 ± 1.67</td><td>20.10 ±0.90</td></tr><tr><td>finetune</td><td>19.75 ± 0.05</td><td>18.55 ± 0.34</td><td>3.53± 0.04</td></tr><tr><td>GEM</td><td>93.25 ±0.36</td><td>24.13 ± 2.46</td><td>11.12 ± 2.48</td></tr><tr><td>iCARL</td><td>83.95 ±0.21</td><td>37.32 ± 2.66</td><td>10.80 ± 0.37</td></tr><tr><td>CURL (Rao et al., 2019)</td><td>92.59 ±0.66</td><td>1</td><td>1</td></tr><tr><td>DN-CPM (Lee et al.,2020)</td><td>93.23±0.09</td><td>45.21 ± 0.18</td><td>20.10 ±0.12</td></tr><tr><td>reservoir</td><td>92.16 ±0.75</td><td>42.48 ± 3.04</td><td>19.57 ± 1.79</td></tr><tr><td>MIR</td><td>93.20±0.36</td><td>42.80 ± 2.22</td><td>20.00±0.57</td></tr><tr><td>GSS</td><td>92.47 ± 0.92</td><td>38.45 ± 1.41</td><td>13.10 ± 0.94</td></tr><tr><td>CoPE-CE</td><td>91.77 ± 0.87</td><td>39.73 ± 2.26</td><td>18.33 ± 1.52</td></tr><tr><td>CoPE (ours)</td><td>93.94± 0.20</td><td>48.92 ± 1.32</td><td>21.62±0.69</td></tr></table>
+
+![](images/2c4551cc46226bfcac2c711518e351365e093a1079d5e77bad2b86df03fbe9ad.jpg)  
+Figure 3: Balanced SplitMNIST first seed $S _ { e v a l }$ t-SNE (Maaten & Hinton, 2008).
+
+![](images/18f24f003d9563b4de9c201fa09f0257e0ad04eb5a52ffbde72e0d15a7939696.jpg)  
+Figure 4: Accuracies over buffer sizes $| { \mathcal { M } } |$ for balanced Split-MNIST and Split-CIFAR10 sequences. The legend reports averages over all observed buffer sizes. $\ast ^ { \ast }$ indicates learner with task information.
+
+# 6 RESULTS AND DISCUSSION
+
+# 6.1 BALANCED DATA STREAMS
+
+The results for the three balanced data streams in Table 1 consistently report state-of-the-art for CoPE. The difficulty for learning online is reflected in the discrepancy of performance between iid-offline and iid-online, indicating increasing difficulty for a minimal $2 \%$ for Split-MNIST, raising by factor 10 for Split-CIFAR10, and culminating to $3 0 \%$ in Split-CIFAR100. For Split-MNIST the gap with iid-online performance is closed by $0 . 7 \%$ compared to main competitors GEM and DN-CPM, with our representations visualized in Figure 3. Furthermore, in the more challenging Split-CIFAR10 setup we significantly increase the gained margin by $3 . 7 \%$ . In the most challenging Split-CIFAR100, CN-DPM, Reservoir and MIR are able to perform on par with the iid-online baseline, however, CoPE establishes an improvement of at least $1 . { \bar { 5 } } \%$ over all four baselines.
+
+Compared to balanced replay with standard cross-entropy (CoPE-CE), the prototypical approach (CoPE) proves effective with significant gains of $2 . 2 \%$ , $9 . 2 \%$ and $3 . 3 \%$ respectively over the three benchmarks. Except for GEM in Split-MNIST, class incremental learning methods GEM and iCaRL are not competing in the online setting and additionally require from the setup to reveal an identifier $t$ to the learner. From the expansion-based methods DN-CPM is competitive, whereas CURL is more suited for unsupervised learning and lacks behind. Although Reservoir and extension MIR perform on par with iid-online for Split-CIFAR100, the imbalanced experiments in Section 6.2 show that full reservoir-based population of the buffer strongly relies on this assumption of equally sized tasks, which is unlikely to occur in real-world data streams.
+
+Buffer size ablation study in Figure 4 shows CoPE to prevail over all sizes of replay buffer $\mathcal { M } _ { r }$ compared to other replay methods, extending robustness to low capacity regimes. Although iCaRL shows competitive results for low capacity, CoPE scales with growing capacity leading to significantly outperforming iCaRL with $1 1 \%$ in Split-MNIST $( 2 k )$ and Split-CIFAR100 $( 5 k )$ , and $1 7 \%$ in SplitCIFAR10 $( 2 k )$ . We refer to Appendix E for the Split-CIFAR100 results.
+
+# 6.2 IMBALANCED DATA STREAMS
+
+Results for the highly imbalanced data stream benchmarks are reported in Figure 4. CoPE significantly outperforms all baselines in the three scenarios, with low standard deviation for the 15 variants indicating robustness over a wide spectrum of imbalanced sequences. Gradient-based sample selection (GSS) outperforms Reservoir and MIR for Split-MNIST, in correspondence with results in Aljundi et al. (2019b), whereas loss-based retrieval in MIR has significant gains for the challenging SplitCIFAR100 setting. However, CoPE surpasses both GSS and MIR for all three benchmarks, and on top of that operates profusely more resource efficient as discussed in Appendix C. The balancing memory scheme in CoPE-CE highly improves Reservoir over imbalanced Split-MNIST and Split-CIFAR10 variants with $1 0 . 8 \%$ and $3 . 4 \%$ respectively, and performs on par for Split-CIFAR100 where balancing over 100 classes with limited batch size proves more difficult. Although CoPE and CoPE-CE share memory and retrieval schemes, the prototypical CoPE surpasses the cross-entropy based CoPE-CE with $4 . 0 \%$ , $2 . 9 \%$ and $6 . 7 \%$ respectively on the three benchmarks, indicating the merits of the PPP-loss and continually evolving prototypes. Figure 6 compares the CoPE and CoPE-CE confusion matrices at the end of learning, showing that CoPE better preserves the recall over early learned classes. CoPE-CE exhibits high plasticity as classes 8 and 9 of the last task have high recall compared to the earlier learned classes. Hence, CoPE seems to better preserve stability, effectively alleviating catastrophic forgetting.
+
+![](images/39b570d5b143fca2b21bf9161153e5bda190f868bfe79da880d4ddbe035fbcfa.jpg)  
+Figure 5: Accuracy $( \% )$ for imbalanced Split-MNIST (left), Split-CIFAR10 (center) and SplitCIFAR100 (right) sequences. The legend reports average accuracies over all the sequence variations.
+
+# 6.3 PPP-LOSS ANALYSIS
+
+In the challenging setting for online processing of non-iid data streams, the PPP-loss exploits information in the small processing batch $B$ , introducing pseudo-prototypes $\hat { \bf p }$ on top of the prototypes. This leads to questioning to what extent the pseudo-prototypes actually contribute to the quality of the embedding, and how this relates to the batch size. We examine both inquiries in Table 2 for the three balanced data streams by comparing inclusion and exclusion of the pseudo-prototypes $\hat { \bf p }$ in the PPP-loss, and extending the batch size $\left| B _ { n } \right|$ . First, including the pseudo-prototypes significantly improves overall performance, and especially for the harder CIFAR-based data streams. Although both setups use batch information to update the prototypes following Eq.(1), it seems crucial to use additional pseudo-prototypes in the PPP-loss to improve latent space quality. Second, results for smaller batch sizes of 10 and 20 are very similar, and deteriorate towards increasing sizes. The PPP-loss implements the expectation over the prototype and the pseudo-prototypes, assuming uniform distribution in Eq.(3). Although this assumption impedes significance of the prototype for increasingly higher batch sizes, it results in ideal robustness for small online processing batches, ideally suited for data incremental learning. Small batches maintain the additional benefit of more frequent prototype updates for the same amount of processed data.
+
+![](images/49034c8b03b406066dce73afc8e44a5b5befd1ae0a9342a51305fd6c0264232b.jpg)  
+Figure 6: CoPE and CoPE-CE confusion matrices at the end of learning averaged over all variations $S ( T _ { i } )$ for the imbalanced Split-MNIST setup in (a) and (b), and Split-CIFAR10 in (c) and (d).
+
+Table 2: Accuracies $( \% )$ for ablating pseudo-prototypes $\hat { \bf p }$ in the PPP-loss and varying batch size.   
+
+<table><tr><td></td><td colspan="2">PPP-loss</td><td colspan="5">Batch Size |Bnl</td></tr><tr><td></td><td>incl. p</td><td>excl. p</td><td>10 (Online)</td><td>20</td><td>50</td><td>100</td><td>200</td></tr><tr><td>Split-MNIST</td><td>93.9± 0.2</td><td>92.4±0.6</td><td>93.9±0.2</td><td>93.9± 0.6</td><td>93.7± 0.3</td><td>93.1±0.6</td><td>89.3± 0.5</td></tr><tr><td>Split-CIFAR10</td><td>48.9 ± 1.3</td><td>41.3 ± 2.0</td><td>48.9 ± 1.3</td><td>48.4± 1.9</td><td>43.4 ± 2.7</td><td>37.4 ± 3.0</td><td>37.0 ± 1.3</td></tr><tr><td>Split-CIFAR100</td><td>21.6 ±0.7</td><td>16.3 ± 0.7</td><td>21.6 ±0.7</td><td>21.7±0.7</td><td>16.5 ± 0.4</td><td>13.8± 0.5</td><td>11.2 ± 0.4</td></tr></table>
+
+# 7 CONCLUSION
+
+In this work, we introduced a new perspective on current paradigms in continual learning with a novel two-agent learner-evaluator framework. To overcome the standard paradigm of static training and testing phases, we explicitly model continual optimization and evaluation in the learner and evaluator agents respectively. We formalized the required resources as the horizon $\mathcal { D }$ , containing the simultaneously available data of the data stream, and the operational memory $\mathcal { M }$ for operation of the learning algorithm. Transitions in the horizon $\mathcal { D } _ { t } \to \mathcal { D } _ { t + 1 }$ enable a uniform differentiation between existing paradigms of task, class and domain incremental learning, and the horizon size encloses the range from online $\mathcal { D } = B$ ) to offline ( $\mathcal { D } = S$ ) learning.
+
+Using the framework, we defined the task-free data incremental learning paradigm, requiring no additional information on the identifier $t$ of the horizon for both the learner and evaluator. In this challenging setup, we proposed Continual Prototype Evolution (CoPE) as a prototypical solution to learn online from non-stationary data streams. As a first, CoPE prevents the prototypes becoming obsolete in an ever evolving representation space, while using the prototypes to combat catastrophic forgetting. The three main components, continually evolving prototypes, a novel Pseudo-Prototypical Proxy loss (PPP-loss), and an efficient balancing replay scheme are proven remarkably effective over 11 baselines in both balanced and highly imbalanced benchmarks. We hope to encourage research in the direction of data incremental learning with online processing of data streams and applications beyond classification.
+
+# REFERENCES
+
+Rahaf Aljundi, Francesca Babiloni, Mohamed Elhoseiny, Marcus Rohrbach, and Tinne Tuytelaars. Memory aware synapses: Learning what (not) to forget. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 139–154, 2018.
+
+Rahaf Aljundi, Eugene Belilovsky, Tinne Tuytelaars, Laurent Charlin, Massimo Caccia, Min Lin, and Lucas Page-Caccia. Online continual learning with maximal interfered retrieval. In Advances in Neural Information Processing Systems, pp. 11849–11860, 2019a.
+
+Rahaf Aljundi, Min Lin, Baptiste Goujaud, and Yoshua Bengio. Gradient based sample selection for online continual learning. In Advances in Neural Information Processing Systems, pp. 11816– 11825, 2019b.
+
+Arindam Banerjee, Srujana Merugu, Inderjit S Dhillon, and Joydeep Ghosh. Clustering with bregman divergences. Journal of machine learning research, 6(Oct):1705–1749, 2005.
+
+Francisco M Castro, Manuel J Marín-Jiménez, Nicolás Guil, Cordelia Schmid, and Karteek Alahari. End-to-end incremental learning. In Proceedings of the European conference on computer vision (ECCV), pp. 233–248, 2018.
+
+Arslan Chaudhry, Marcus Rohrbach, Mohamed Elhoseiny, Thalaiyasingam Ajanthan, Puneet K Dokania, Philip HS Torr, and Marc’Aurelio Ranzato. Continual learning with tiny episodic memories. arXiv preprint arXiv:1902.10486, 2019.
+
+Aristotelis Chrysakis and Marie-Francine Moens. Online continual learning from imbalanced data. Proceedings of Machine Learning Research, 2020.
+
+Matthias De Lange, Rahaf Aljundi, Marc Masana, Sarah Parisot, Xu Jia, Ales Leonardis, Gregory Slabaugh, and Tinne Tuytelaars. A continual learning survey: Defying forgetting in classification tasks. arXiv preprint arXiv:1909.08383, 2019.
+
+Matthias De Lange, Xu Jia, Sarah Parisot, Ales Leonardis, Gregory Slabaugh, and Tinne Tuytelaars. Unsupervised model personalization while preserving privacy and scalability: An open problem. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 14463–14472, 2020.
+
+Robert M French. Catastrophic forgetting in connectionist networks. Trends in cognitive sciences, 3 (4):128–135, 1999.
+
+João Gama, Indre Žliobait ˙ e, Albert Bifet, Mykola Pechenizkiy, and Abdelhamid Bouchachia. A ˙ survey on concept drift adaptation. ACM computing surveys (CSUR), 46(4):1–37, 2014.
+
+Ian J Goodfellow, Mehdi Mirza, Da Xiao, Aaron Courville, and Yoshua Bengio. An empirical investigation of catastrophic forgetting in gradient-based neural networks. arXiv preprint arXiv:1312.6211, 2013.
+
+Stephen Grossberg. Studies of mind and brain : neural principles of learning, perception, development, cognition, and motor control. Boston studies in the philosophy of science 70. Reidel, Dordrecht, 1982. ISBN 9027713596.
+
+Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), volume 2, pp. 1735–1742. IEEE, 2006.
+
+Xu Han, Yi Dai, Tianyu Gao, Yankai Lin, Zhiyuan Liu, Peng Li, Maosong Sun, and Jie Zhou. Continual relation learning via episodic memory activation and reconsolidation. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pp. 6429–6440, 2020.
+
+Ben Harwood, BG Kumar, Gustavo Carneiro, Ian Reid, Tom Drummond, et al. Smart mining for deep metric learning. In Proceedings of the IEEE International Conference on Computer Vision, pp. 2821–2829, 2017.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.
+
+Yen-Chang Hsu, Yen-Cheng Liu, Anita Ramasamy, and Zsolt Kira. Re-evaluating continual learning scenarios: A categorization and case for strong baselines. arXiv preprint arXiv:1810.12488, 2018.
+
+Daniel Kahneman and Amos Tversky. Subjective probability: A judgment of representativeness. Cognitive psychology, 3(3):430–454, 1972.
+
+Nitin Kamra, Umang Gupta, and Yan Liu. Deep generative dual memory network for continual learning. arXiv preprint arXiv:1710.10368, 2017.
+
+Chris Dongjoo Kim, Jinseo Jeong, and Gunhee Kim. Imbalanced continual learning with partitioning reservoir sampling. arXiv preprint arXiv:2009.03632, 2020.
+
+James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, et al. Overcoming catastrophic forgetting in neural networks. Proceedings of the national academy of sciences, 114 (13):3521–3526, 2017.
+
+Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.
+
+Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Soochan Lee, Junsoo Ha, Dongsu Zhang, and Gunhee Kim. A neural dirichlet process mixture model for task-free continual learning. arXiv preprint arXiv:2001.00689, 2020.
+
+Zhizhong Li and Derek Hoiem. Learning without forgetting. IEEE transactions on pattern analysis and machine intelligence, 40(12):2935–2947, 2017.
+
+David Lopez-Paz and Marc’Aurelio Ranzato. Gradient episodic memory for continual learning. In Advances in Neural Information Processing Systems, pp. 6467–6476, 2017.
+
+Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(Nov):2579–2605, 2008.
+
+Arun Mallya and Svetlana Lazebnik. Packnet: Adding multiple tasks to a single network by iterative pruning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 7765–7773, 2018.
+
+Thomas Mensink, Jakob Verbeek, Florent Perronnin, and Gabriela Csurka. Distance-based image classification: Generalizing to new classes at near-zero cost. IEEE transactions on pattern analysis and machine intelligence, 35(11):2624–2637, 2013.
+
+Hyun Oh Song, Yu Xiang, Stefanie Jegelka, and Silvio Savarese. Deep metric learning via lifted structured feature embedding. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4004–4012, 2016.
+
+Dushyant Rao, Francesco Visin, Andrei Rusu, Razvan Pascanu, Yee Whye Teh, and Raia Hadsell. Continual unsupervised representation learning. In Advances in Neural Information Processing Systems, pp. 7645–7655, 2019.
+
+Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, Georg Sperl, and Christoph H Lampert. icarl: Incremental classifier and representation learning. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pp. 2001–2010, 2017.
+
+Jeffrey C Schlimmer and Richard H Granger. Beyond incremental processing: Tracking concept drift. In AAAI, pp. 502–507, 1986.
+
+Ari Seff, Alex Beatson, Daniel Suo, and Han Liu. Continual learning in generative adversarial nets. arXiv preprint arXiv:1705.08395, 2017.   
+Joan Serra, Didac Suris, Marius Miron, and Alexandros Karatzoglou. Overcoming catastrophic forgetting with hard attention to the task. arXiv preprint arXiv:1801.01423, 2018.   
+Hanul Shin, Jung Kwon Lee, Jaehong Kim, and Jiwon Kim. Continual learning with deep generative replay. In Advances in Neural Information Processing Systems, pp. 2990–2999, 2017.   
+Konstantin Shmelkov, Cordelia Schmid, and Karteek Alahari. Incremental learning of object detectors without catastrophic forgetting. In Proceedings of the IEEE International Conference on Computer Vision, pp. 3400–3409, 2017.   
+Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Advances in neural information processing systems, pp. 4077–4087, 2017.   
+Alexey Tsymbal. The problem of concept drift: definitions and related work. Computer Science Department, Trinity College Dublin, 106(2):58, 2004.   
+Gido M van de Ven and Andreas S Tolias. Generative replay with feedback connections as a general strategy for continual learning. arXiv preprint arXiv:1809.10635, 2018.   
+Gido M van de Ven and Andreas S Tolias. Three scenarios for continual learning. arXiv preprint arXiv:1904.07734, 2019.   
+Jeffrey S Vitter. Random sampling with a reservoir. ACM Transactions on Mathematical Software (TOMS), 11(1):37–57, 1985.   
+Yue Wu, Yinpeng Chen, Lijuan Wang, Yuancheng Ye, Zicheng Liu, Yandong Guo, Zhengyou Zhang, and Yun Fu. Incremental classifier learning with generative adversarial networks. arXiv preprint arXiv:1802.00853, 2018.   
+Mang Ye, Xu Zhang, Pong C Yuen, and Shih-Fu Chang. Unsupervised embedding learning via invariant and spreading instance feature. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 6210–6219, 2019.   
+Friedemann Zenke, Ben Poole, and Surya Ganguli. Continual learning through synaptic intelligence. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 3987– 3995. JMLR. org, 2017.
+
+# APPENDIX
+
+# A ALGORITHM
+
+Our proposed algorithm is fully formalized in this section, as well as in our code that will be made publicly available on acceptance of this paper. Algorithm 1 and Algorithm 2 describe the learner for CoPE, whereas the evaluator uses $c ^ { * } = { \arg \operatorname* { m a x } _ { c \in \mathcal { Y } } \mathbf { f } _ { i } ^ { T } \ \mathbf { p } ^ { c } }$ , classifying $\mathbf { x } _ { i }$ as category $c ^ { * }$ with the most similar prototype $\mathbf { p } ^ { c ^ { * } }$ . As for a true continually progressing system, the evaluator can urge prediction at any point in time, while the learner keeps acquiring knowledge from the data stream.
+
+Algorithm 1 The CoPE learner in the data incremental learning setup.   
+Require: data stream $S$ , prototype momentum $\alpha$ , memory capacity $M$ , learning rate $\eta$   
+Initialize operational memory $\mathcal { M } = \emptyset$ , observed classes $\mathscr { y } = \emptyset$ , sample count per class $N = \emptyset$ ,   
+model parameters $\theta$   
+1: for $\bar { B _ { n } } \overset { \cdot } { = } \{ ( \mathbf { x } _ { 1 } , \mathbf { y } _ { 1 } ) , . . . , ( \mathbf { x } _ { | B _ { n } | } , \mathbf { y } _ { | B _ { n } | } ) \} \sim S$ do $\triangleright$ Data stream batch w/o task information   
+2: $B _ { \mathcal { M } } \gets \mathrm { R A N D O M S A M P L E } ( \mathcal { M } _ { r } , | B _ { n } | )$ $\triangleright$ Randomly sample $\left| B _ { n } \right|$ exemplars from $\mathcal { M } _ { r }$   
+3: $B = \varnothing$   
+4: for $( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \in B _ { n } \cup B _ { \mathcal { M } }$ do   
+5: if $y _ { i } \notin \mathcal { V }$ then   
+6: INITCLASS $( \mathcal { M } , N , \mathcal { V } , y _ { i } )$ . Initialize memory and prototype   
+7: end if   
+8: $B  B \cup f _ { \theta } ( \mathbf { x } _ { i } )$ . Collect features   
+9: end for   
+10: $\mathcal { L }  0$ . Initialize loss   
+11: for $\mathbf { f } _ { i } ^ { c } \in \mathcal { B }$ do   
+12: $\begin{array} { r l r } { \dot { \mathcal { L } } \gets \mathcal { L } - \frac { 1 } { | \mathcal { B } | } \left[ \log P ( c | { \bf x } _ { i } ^ { c } ) + \sum _ { { \bf x } _ { j } ^ { k } } \log ( 1 - P ( c | { \bf x } _ { j } ^ { k } ) ) \right] } & { } & { \mathrm { s } \operatorname { s u m } \mathrm { a l } \mathrm { i n s t a n c e s } \mathrm { P P P - l o s s } } \end{array}$   
+13: end for   
+14: θ θ + η . Optimize objective with SGD   
+15: $\mathbf { P R O T O T Y P E U P D A T E } ( \mathcal { M } _ { p } , \ B , N , \alpha )$ . Update prototypes in $\mathcal { M } _ { p }$   
+16: MEMORYUPDATE $( \mathcal { M } _ { r } , B _ { n } , N )$ . Update memory $\mathcal { M } _ { r }$ with new input samples   
+17: end for
+
+Algorithm 2 Memory Management of the replay memory and prototypes. UNIFORMRd( samples elements in a $d$ -dimensional vector with uniform probability in range $[ s _ { 1 } , s _ { 2 } ] \in \mathbb { R }$ .   
+
+<table><tr><td colspan="3">sampies elements ina a-dimensional VectorWitn unilormprobabiityinrange[Si,S2]∈R. Require: memory capacity M 1: function PROTOTYPEUPDATE(Mp, B, N, α)</td></tr><tr><td>1: function INITCLAss(M,N,V, y)</td><td></td><td>2: for pc ∈ Mp do</td></tr><tr><td>2:</td><td>N ← NU{Ny =O} Sample counts</td><td>3: Nc ← Nc+|Bc</td></tr><tr><td>3: y↑yu{}</td><td>Observed classes</td><td>p =Bq∑feeBe fc 4:</td></tr><tr><td>4: m = M/||</td><td>Capacity per class 5:</td><td>p←ap+(1-a)pc</td></tr><tr><td>5:</td><td>for M= (x1,., X|M|) ∈Mr do 6:</td><td>p←p/pll2 Normalize</td></tr><tr><td>6: end for</td><td>M ←(x1,., Xm）&gt; Keep first m 7:</td><td>end for</td></tr><tr><td>7:</td><td></td><td>8: end function</td></tr><tr><td>8:</td><td>M←MU{My =0}</td><td>9: function MEMORYUPDATE(Mr,Bn, N)</td></tr><tr><td>9:</td><td>p ←UNIFORMd(0,1) 10:</td><td>for x ∈ Bn do Class Reservoir</td></tr><tr><td>10: 11:</td><td>My←{p²/|p|l2} Init prototype 11:</td><td>j = UNIFORMN1 (1, Nc)</td></tr><tr><td>end function</td><td>12:</td><td>if j≤|M| then</td></tr><tr><td></td><td>13:</td><td>M [j] ← x Replace exemplar</td></tr><tr><td></td><td>14:</td><td>end if</td></tr><tr><td></td><td>15:</td><td>end for</td></tr><tr><td></td><td>16:</td><td></td></tr><tr><td></td><td></td><td>end function</td></tr></table>
+
+# B SETUP
+
+A gridsearch in the online continual learning setup was adopted, selecting the setup with highest performance, similar to (Lopez-Paz & Ranzato, 2017). All methods are prone to learning rate gridsearch $[ 0 . 0 5 , 0 . 0 1 , 0 . 0 0 5 , 0 . 0 0 1 ]$ . iCaRL knowledge distillation strength is set to 1, and GEM bias is set to 0.5, following (Lopez-Paz & Ranzato, 2017; Rebuffi et al., 2017; Aljundi et al., 2019b). GSS and MIR follow their original setup from their codebase in (Aljundi et al., 2019b) and (Aljundi et al., 2019a), with our additional learning rate gridsearch. CURL (Rao et al., 2019) and DN-CPM (Lee et al., 2020) results, and the best imbalanced Split-MNIST results out of the greedy/IQP versions for GSS (Aljundi et al., 2019b) are reported from their original works. CoPE searched for a suitable temperature $\tau = [ 0 . 1 , 0 . 2 , . . . , 1 , 2 ]$ which was set to 0.1 for all balanced and imbalanced SplitMNIST and Split-CIFAR10 experiments, similar to (Ye et al., 2019). Based on the ablation study in Appendix D, we set the prototypical momentum fixed to 0.99. For the challenging Split-CIFAR100 setting methods are allowed multiple iterations per batch as in (Lopez-Paz & Ranzato, 2017), from which the best results are selected (baselines, reservoir, CN-DPM perform 1 iteration, others 5). The CIFAR100 temperature required higher concentration with $\tau = 0 . 0 5$ and prototypical momentum 0.9. For the balanced setups, the latent dimensionality $d$ is fixed to 100 for Split-MNIST as in (Rao et al., 2019), and selected 256 in a gridsearch [128, 256] and [128, 256, 512] for Split-CIFAR10 and Split-CIFAR100 respectively. The imbalanced benchmarks follow the low capacity setup in Appendix E.1, with $d \in [ 1 6 , 3 2 , 6 4 ]$ set to 64 for Split-MNIST and $d \in [ 1 2 8 , 2 5 6 ]$ set to 128 for Split-CIFAR10 and 256 for Split-CIFAR100. Results are obtained without L2 normalization of the prototypes as we found it to have insignificant effect. The CIFAR10 labels in the confusion matrices from 0 to 10 stand for the indices in the following list: [airplane, automobile, bird, cat, deer, dog, frog, horse, ship, truck]. We will make our code publicly available upon acceptance of this paper to ensure reproducibility.
+
+# C RESOURCE ANALYSIS TASK-FREE REPLAY METHODS
+
+In this section we compare usage of computational and memory resources for the replay methods fitted for the online data incremental learning paradigm.
+
+Reservoir is a powerful baseline for balanced data streams (Chaudhry et al., 2019), with only minimal computational cost by keeping count $n$ of how many samples have been observed. This count is then used relative to the buffer size $M$ to define the probability $M / n$ to store the new sample. As shown in the imbalanced data stream experiments, Reservoir is not fit for more real-world scenarios with typically varying frequency of occurrence per class. Improving this simple experience replay has led to research focusing on more complex strategies, discussed in the following.
+
+MIR (Aljundi et al., 2019a) replaces the random retrieval from the buffer in Reservoir with a lossbased approach. They store a momentary update of the network optimized for the new incoming batch and calculate the change in loss for a random subset of replay memories $\tilde { B }$ , which is larger than the batch size (ideally five times the batch size for their experiments (Aljundi et al., 2019a)). Besides a copy of the full model, this also requires calculating the loss twice in a sequential manner for the full subset $\tilde { B }$ and an extra temporary model update using only the new batch $B _ { n }$ , both significantly increasing processing time for the learner.
+
+GSS (Aljundi et al., 2019b) resides with Reservoir to use random retrieval of the buffer, but proposes a gradient-based population strategy. They introduce two variants, in which the first solves an Integer Quadratic Problem (IQP) with polynomial complexity w.r.t. the replay memory. As this is not scalable, they also propose a stochastic GSS-greedy variant. This more efficient GSS-greedy approach requires an additional forward pass, loss calculation, and backwards pass to obtain the gradients for the full considered subset $\tilde { B }$ in the memory. Additionally, it uses similarities of the gradients for stochastic sample selection in the replay memory $\mathcal { M } _ { r }$ , straining memory requirements as batch $B _ { n }$ requires for each sample $| \tilde { B } | + 1$ gradients to be accessed simultaneously to calculate $| \tilde { B } |$ cosine similarities in the high-dimensional gradient-space.
+
+CoPE (ours) resembles Reservoir’s memory population by keeping count of the samples per classspecific replay memory subset. The PPP-loss requires calculation of a similarity matrix with all the $d$ -dimensional representations in the batch $B$ . Using a normalized cosine similarity, this implies efficient matrix multiplication with the low-dimensional vectors. This is in high contrast to GSS, which calculates cosine similarity in the full high-dimensional gradient space for additional samples that are not present in current batch $B$ , and therefore requires additional costly forward and backward passes. Furthermore, in our prototypical approach the prototype momentum updates also rely solely on samples that are in the current batch $B$ , hence requiring only minimal additional computation. Comparing to both MIR and GSS, we don’t require storing model copies or additional gradients, but merely store low-dimensional prototypes for each class, saving a significant amount of required storage space. For example, a Resnet18 model requires 11.7 million parameters to enable model copies or gradients, whereas our method even for 1000-way classification with $d = 1 0 2 4$ would require only $9 \%$ of the model capacity in memory for the prototypes.
+
+# D EXTENDED ABLATION STUDY
+
+# D.1 ABLATION PROTOTYPE MOMENTUM
+
+In all experiments, a high momentum is employed to update prototypes with the latent mean of the batch. Table 3 illustrates the influence of higher momentum $( \geq 0 . 9 )$ . Compared to low momentum of 0.1, Split-MNIST only gains a small margin of $0 . 4 5 \%$ , whereas Split-CIFAR10 and Split-CIFAR100 significantly improve with at least $3 . 0 \%$ and $4 . 2 \%$ respectively. Using momentum prevents the prototype to rely solely on the current batch instances, and higher momentum values attain a more gradual change of the prototypes by stabilizing its trajectory in the ever-evolving latent space.
+
+Table 3: Ablation study changing momentum strength for prototype updates, reported in average accuracy $( \% )$ over 5 runs. Higher momentum values $( \geq 0 . 9 )$ obtain better performance, especially for the CIFAR sequences, compared to low momentum (0.1).   
+
+<table><tr><td></td><td colspan="4">Prototype Momentum</td></tr><tr><td></td><td>0.1</td><td>0.9</td><td>0.95</td><td>0.99</td></tr><tr><td>Split-MNIST</td><td>93.49 ± 0.70</td><td>94.11 ± 0.34</td><td>93.96 ± 0.30</td><td>93.94 ± 0.20</td></tr><tr><td>Split-CIFAR10</td><td>44.48 ± 3.19</td><td>48.02 ± 2.49</td><td>47.98 ± 3.14</td><td>48.92 ± 1.32</td></tr><tr><td>Split-CIFAR100</td><td>15.79 ± 1.16</td><td>21.62 ± 0.69</td><td>21.56 ± 0.58</td><td>20.01 ± 1.81</td></tr></table>
+
+# D.2 ABLATION INTER AND INTRA-CLASS VARIANCE TERMS PPP-LOSS
+
+In this section, the importance is scrutinized of the two loss components to enhance inter and intraclass variance in the PPP-loss. Table 4 compares using only positive pairs from the batch in the attractor $( \mathcal { L } _ { p o s } )$ or only negative pairs in the repellor $( \mathcal { L } _ { n e g } )$ to the full-fledged PPP-loss $( \mathcal { L } )$ . The attractor term shows competitive performance to the full PPP-loss for Split-MNIST, but deteriorates as the data streams become harder for the CIFAR setups. The repellor term is on par with the full PPP-loss for Split-MNIST and Split-CIFAR10, but collapses for Split-CIFAR100. The latter is challenging due to the high number of classes with only a batch size of 10, which impedes having pseudo-prototypes of all classes in the same batch. The PPP-loss incorporates both reduction of intra-class variance with the attractor term and increases inter-class variance with the repellor term, attaining state-of-the-art performance.
+
+Besides isolating the attractor and repellor terms of the PPP-loss in the ablation study, we further investigate the weighing of the two terms during the lifetime of the learner in Figure 7. We average results over 5 runs for balanced Split-MNIST, finding the repellor to dominate. This trend is to be expected as the repellor term in Eq.(5) has per instance a summation over all other class instances. The attractor term has minimal influence especially for data presented for the first task. This indicates the samples in the binary latent space (having observed only two classes) majorly repelling rather than attracting samples. The embedding network is still learning the initial features, and overlap in the two latent class distributions summed over the other class samples results in a prevailing repellor term.
+
+![](images/d696885057dc1678c89b7685eda3ea4be239fb7205ca3450b46207a2c9f4b6a9.jpg)  
+Figure 7: Weighing $( \% )$ between the positive loss term $\mathcal { L } _ { p o s }$ compared to the full PPP-loss $\mathcal { L }$ averaged over 5 runs of balanced Split-MNIST with standard deviation in blue.
+
+# D.3 PSEUDO-PROTOTYPE ABLATION VISUALIZATION
+
+In the main paper we find in an ablation study that using pseudo-prototypes $\hat { \bf p }$ as proxy for the class-mean has significant improvements for the PPP-loss. Additionally, Figure 8 shows this in a 2-dimensional t-SNE space for the first seed of the balanced Split-MNIST experiment. Including the pseudo-prototypes (incl. pˆ) illustrates a striking degree of inter-class variance in Figure 8a, whereas more interference occurs when excluding the pseudo-prototypes in Figure 8b. This is reflected in the performance, as including prototypes results in $9 4 . 5 2 \%$ accuracy, whereas excluding them has only $\bar { 9 0 . 8 6 \% }$ for the first seed.
+
+![](images/b6141f3dfa4d02c6733abd460c81ac774161c82551611d2b58c33953d354cd79.jpg)  
+Figure 8: Split-MNIST first seed t-SNE representation of the test data $S _ { e v a l }$ , including (a) and excluding (b) the pseudo-prototypes $\hat { \bf p }$ in the PPP-loss.
+
+# E ADDITIONAL EXPERIMENTS
+
+# E.1 BALANCED DATA STREAMS WITH LOW CAPACITY
+
+In these experiments we scrutinize performance of CoPE with less capacity in the memory and model, and with shorter data streams. All methods are allowed multiple iterations (maximal 5) as in (Aljundi et al., 2019b). Results are averaged over 5 seeds. Similar to the setup of GSS (Aljundi et al., 2019b), we adopt two data sequences with truncated data per task:
+
+• Split-MNIST-mini is similar to the Split-MNIST data stream with 5 tasks, but each task is confined to 1k training samples. Evaluation considers the full test subset. The network is an MLP with two hidden layers of 100 units, with total memory size of $0 . 3 \mathrm { k }$ exemplars. Latent dimensionality $d$ is selected 32 from [16, 32, 64].   
+Split-CIFAR10-mini is similar to the Split-CIFAR10 data stream with 5 tasks, but each task comprises $2 \mathrm { k }$ training samples, with a total subset of 10k samples out of the $5 0 \mathrm { k }$ available. The full test subset is used for evaluation. The network used is the same ResNet18 as in the main paper, with total memory size of 1k exemplars. Latent dimensionality $d$ is selected 128 from [128, 256].
+
+Analysis. Table 5 shows the results for Split-MNIST-mini and Split-CIFAR10-mini, with GSS and DN-CPM results reported from their original works in a corresponding setup. In Split-MNIST-mini our method approaches the iid-online baseline up to $1 \%$ , and outperforms its closest competitors GEM and MIR with at least $1 . 4 5 \%$ . In Split-CIFAR10-mini CoPE saliently surpasses the iid-online baseline with $2 . 2 5 \%$ , hence outperforming online training over an iid datastream. Moreover, CoPE surpasses CN-DPM by $3 \%$ . Reservoir proves a strong baseline, with in this case the additional MIR loss-based retrieval decreasing performance. Similar to our findings in the main paper and Aljundi et al. (2019a;b), GEM encounters difficulties in a CIFAR10 based setup, for which we find the bias hyperparameter $\gamma \geq 0$ in the gradient projection to have insignificant influence. These results confirm CoPE outperforming both GSS and CN-DPM in this low capacity setting established in their original work.
+
+Table 5: Split-MNIST-mini and Split-CIFAR10-mini results, with respectively only 1k and 2k samples per task. GSS and DN-CPM results reported from original work in these setups.   
+
+<table><tr><td></td><td>Split-MNIST-mini</td><td>Split-CIFAR10-mini</td></tr><tr><td>iid-offline</td><td>94.58 ± 0.17</td><td>67.41 ± 1.37</td></tr><tr><td>iid-online</td><td>87.57 ± 3.54</td><td>42.50 ± 2.15</td></tr><tr><td>finetune</td><td>21.74 ± 3.38</td><td>16.65 ± 0.24</td></tr><tr><td>GEM</td><td>85.09 ± 0.52</td><td>22.31 ± 1.37</td></tr><tr><td>iCaRL</td><td>83.23 ± 0.92</td><td>26.54 ± 2.73</td></tr><tr><td>DN-CPM (Lee et al., 2020)</td><td></td><td>41.78</td></tr><tr><td>reservoir</td><td>82.73 ± 2.39</td><td>38.21 ± 3.39</td></tr><tr><td>MIR</td><td>84.40 ± 0.91</td><td>37.20 ± 2.74</td></tr><tr><td>GSS (Aljundi et al., 2019b)</td><td>82.60 ± 2.90</td><td>33.56 ± 1.70</td></tr><tr><td>CoPE</td><td>86.54± 1.41</td><td>44.75 ± 2.68</td></tr></table>
+
+# E.2 BUFFER SIZE ANALYSIS: SPLIT-CIFAR100
+
+The results for Split-CIFAR10 and Split-MNIST are reported in the main paper, whereas SplitCIFAR100 results are added here in Figure 9 due to lack of space. We observe the same trend, where CoPE prevails over other replay methods by high margin from low to high-capacity regimes. The performance of the learner in CoPE scales with the size of $\mathcal { M } _ { r }$ .
+
+# E.3 UNABALANCED BENCHMARK RESULTS
+
+The graphs in the main paper visualize the numbers in Table 6, which we fully report here as a reference for future work. Each $S ( T _ { i } )$ data stream performance is averaged over five different initial seeds. The ’Avg.’ results average over all mean performances of the dataset variants $S ( T _ { i } )$ .
+
+![](images/fd6c820c82ac2b0d4c0647a932919e44e48e3da0e015fff65e4add08eb7af74a.jpg)  
+Figure 9: Accuracies over buffer sizes $| { \mathcal { M } } |$ for balanced Split-CIFAR100 sequence.
+
+Table 6: Numeric results for imbalanced Split-MNIST, Split-CIFAR10 and Split-CIFAR100 sequences.   
+
+<table><tr><td>Dataset</td><td>Imbalanced Sequence</td><td>CoPE</td><td>CoPE-CE</td><td>GSS</td><td>MIR</td><td>Reservoir</td></tr><tr><td>Split-MNIST</td><td>S(Ti）</td><td>83.4± 2.0</td><td>81.8 ± 1.2</td><td>75.9 ± 3.2</td><td>64.8± 5.1</td><td>64.2 ± 2.3</td></tr><tr><td></td><td>S（T）</td><td>84.5 ± 1.6</td><td>80.1 ± 1.9</td><td>78.5± 2.7</td><td>67.4 ± 3.2</td><td>65.5 ± 4.6</td></tr><tr><td></td><td>S(T）</td><td>85.1± 0.6</td><td>79.6 ± 2.0</td><td>81.5 ± 2.3</td><td>72.4 ± 3.0</td><td>72.1± 4.0</td></tr><tr><td></td><td>S(T4）</td><td>84.8± 1.0</td><td>80.0 ± 3.1</td><td>79.5 ± 0.6</td><td>72.6 ± 3.1</td><td>73.6 ± 2.4</td></tr><tr><td></td><td>S(T）</td><td>84.0 ± 1.3</td><td>80.7 ± 1.8</td><td>79.1± 0.7</td><td>77.2 ± 3.4</td><td>73.2 ± 4.0</td></tr><tr><td></td><td>Avg.</td><td>84.4±0.7</td><td>80.4± 0.9</td><td>78.9 ± 2.0</td><td>70.9 ± 4.9</td><td>69.7 ± 4.5</td></tr><tr><td>Split-CIFAR10</td><td>S(T1)</td><td>39.0 ± 1.3</td><td>36.4 ± 3.0</td><td>32.3 ± 3.0</td><td>32.6 ± 3.6</td><td>35.5 ± 3.4</td></tr><tr><td></td><td>S（T）</td><td>35.3 ± 2.6</td><td>34.1 ± 2.8</td><td>28.3 ± 0.4</td><td>27.2 ± 1.8</td><td>29.3± 2.8</td></tr><tr><td></td><td>S（T）</td><td>36.2 ± 2.5</td><td>34.6 ± 2.5</td><td>29.5 ± 1.5</td><td>29.6 ± 2.1</td><td>31.4± 2.1</td></tr><tr><td></td><td>S(T4）</td><td>39.1 ± 2.4</td><td>33.5 ± 4.2</td><td>34.6 ± 1.3</td><td>31.0 ± 2.3</td><td>32.1± 0.6</td></tr><tr><td></td><td>S(T)</td><td>37.3 ± 3.3</td><td>33.9 ± 2.9</td><td>28.3±2.4</td><td>27.6 ± 2.7</td><td>28.8± 1.9</td></tr><tr><td></td><td>Avg.</td><td>37.4 ± 1.7</td><td>34.5 ± 1.1</td><td>30.6 ± 2.8</td><td>29.6 ± 2.3</td><td>31.4 ± 2.7</td></tr><tr><td>Split-CIFAR100</td><td>S(Ti)</td><td>18.2 ± 0.6</td><td>11.7 ± 0.6</td><td>10.2 ± 0.8</td><td>18.4± 0.9</td><td>11.1± 0.6</td></tr><tr><td></td><td>S(T）</td><td>18.5 ± 1.3</td><td>12.6 ± 1.2</td><td>10.7 ± 0.5</td><td>17.6 ± 0.9</td><td>11.5 ± 1.4</td></tr><tr><td></td><td>S（Ti0）</td><td>19.2 ± 0.9</td><td>11.1 ± 0.7</td><td>11.1 ± 0.3</td><td>17.8 ± 0.7</td><td>11.9 ± 0.7</td></tr><tr><td></td><td>S(T15)</td><td>18.7 ± 0.6</td><td>11.2 ± 0.8</td><td>11.1 ± 0.9</td><td>17.8 ± 0.9</td><td>12.1 ± 0.8</td></tr><tr><td></td><td>S(T20)</td><td>18.5 ± 1.5</td><td>12.8 ± 1.3</td><td>11.1 ± 0.4</td><td>17.6 ± 0.4</td><td>12.5 ± 1.1</td></tr><tr><td></td><td>Avg.</td><td>18.6 ± 0.4</td><td>11.9 ± 0.8</td><td>10.8± 0.4</td><td>17.8 ± 0.3</td><td>11.8 ± 0.5</td></tr></table>

parse/train/V1N4GEWki_E/V1N4GEWki_E.md

@@ -0,0 +1,382 @@
+# GRADIENT FLOW IN SPARSE NEURAL NETWORKS AND HOW LOTTERY TICKETS WIN
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Sparse Neural Networks (NNs) can match the generalization of dense NNs using a fraction of the compute/storage for inference, and have the potential to enable efficient training. However, naively training unstructured sparse NNs from random initialization results in significantly worse generalization, with the notable exceptions of Lottery Tickets (LTs) and Dynamic Sparse Training (DST). In this work, we attempt to answer: (1) why training unstructured sparse networks from random initialization performs poorly and; and (2) what makes LTs and DST the exceptions? We show that sparse NNs have poor gradient flow at initialization and propose a modified initialization for unstructured connectivity. Furthermore, we find that DST methods significantly improve gradient flow during training over traditional sparse training methods. Finally, we show that LTs do not improve gradient flow, rather their success lies in re-learning the pruning solution they are derived from — however, this comes at the cost of learning novel solutions.
+
+# 1 Introduction
+
+Deep Neural Networks (DNNs) are the state-of-the-art method for solving problems in computer vision, speech recognition, and many other fields. While early research in deep learning focused on application to new problems, or pushing state-of-the-art performance with ever larger/more computationally expensive models, a broader focus has emerged towards their efficient real-world application. One such focus is on the observation that only a sparse subset of this dense connectivity is required for inference, as apparent in the success of pruning (Han et al., 2015; Mozer et al., 1989b).
+
+Pruning has a long history in Neural Network (NN) literature, and remains the most popular approach for finding sparse NNs. Sparse NNs found by pruning algorithms (Han et al., 2015; Louizos et al., 2017; Molchanov et al., 2017; Zhu et al., 2018) (i.e. pruning solutions) can match dense NN generalization with much better efficiency at inference time. However, naively training an (unstructured) sparse NN from a random initialization (i.e. from scratch), typically leads to significantly worse generalization.
+
+Two methods in particular have shown some success at addressing this problem — Lottery Tickets (LTs) and Dynamic Sparse Training (DST). The mechanism behind the success of both of these methods is not well understood however, e.g. we don’t know how to find Lottery Tickets (LTs) efficiently; while RigL (Evci et al., 2020), a recent DST method, requires $5 \times$ the training steps to match dense NN generalization. Only in understanding how these methods overcome the difficulty of sparse training can we improve upon them.
+
+A significant breakthrough in training DNNs — addressing vanishing and exploding gradients — arose from understanding gradient flow both at initialization, and during training. In this work we investigate the role of gradient flow in the difficulty of training unstructured sparse NNs from random initializations and from LT initializations. Our experimental investigation results in the following insights:
+
+1. Sparse NNs have poor gradient flow at initialization. In $\ S 3 . 1$ , $\ S 4 . 1$ we show that existing methods for initializing sparse NNs are incorrect in not considering heterogeneous connectivity. We believe we are the first to show that sparsity-aware initialization methods improve gradient flow and training. 2. Sparse NNs have poor gradient flow during training. In $\ S 3 . 2$ , $\ S 4 . 2$ , we observe that even in sparse NN architectures less sensitive to incorrect initialization, the gradient flow during training is poor. We show that DST methods achieving the best generalization have improved gradient flow. 3. Lottery Tickets don’t improve upon (1) or (2), instead they re-learn the pruning solution. In $\ S 3 . 3$ , $\ S 4 . 3$ we show that a LT initialization resides within the same basin of attraction as the original pruning solution it is derived of, and a LT solution is highly similar to the pruning solution in function space.
+
+# 2 Related Work
+
+Pruning Pruning is used commonly in Neural Network (NN) literature to obtain sparse networks (Castellano et al., 1997; Hanson et al., 1988; Kusupati et al., 2020; Mozer et al., 1989a,b; Setiono, 1997; Sietsma et al., 1988; Wortsman et al., 2019). Pruning algorithms remove connections of a trained dense network using various criteria including weight magnitude (Han et al., 2016, 2015; Zhu et al., 2018), gradient-based measures (Molchanov et al., 2016), and $2 ^ { \mathrm { n d } }$ -order terms based on the Hessian (Hassibi et al., 1993; LeCun et al., 1990). While the majority of pruning algorithms focus on pruning after training, a subset focuses on pruning NNs before training (Lee et al., 2019; Tanaka et al., 2020; Wang et al., 2020). Gradient Signal Preservation (GRaSP) (Wang et al., 2020) is particularly relevant to our study, since their pruning criteria aims to preserve gradient flow, and they observe a positive correlation between initial gradient flow and final generalization. However, recent work of Frankle et al., 2020b suggests that the reported gains are due to sparsity distributions discovered rather than the particular sub-network. Another limitation of these algorithms is that they don’t scale to large scale tasks like Resnet-50 training on ImageNet-2012.
+
+Lottery Tickets Frankle et al. (2019a) showed the existence of sparse sub-networks at initialization — known as Lottery Tickets — which can be trained to match the generalization of the corresponding dense Deep Neural Network (DNN). The initial work of Frankle et al. (2019a) inspired much follow-up work. Gale et al. (2019) and Liu et al. (2019) observed that the initial formulation was not applicable to larger networks with higher learning rates. Frankle et al. (2019b, 2020a) proposed late rewinding as a solution. Morcos et al. (2019) and Sabatelli et al. (2020) showed that Lottery Tickets (LTs) trained on large datasets transfer to smaller ones, but not vice versa. Frankle et al. (2020c), Ramanujan et al. (2019), and Zhou et al. (2019) focused on further understanding LTs, and finding sparse sub-networks at initialization. As one might expect, sufficiently large networks would have smaller solutions hidden in them. Malach et al. (2020) studied this and proved the existence of solutions in sufficiently large networks. However, it is an open question whether finding such networks at initialization could be done more efficiently than with existing pruning algorithms.
+
+Dynamic Sparse Training Most training algorithms work on pre-determined architectures and optimize parameters using fixed learning schedules. Dynamic Sparse Training (DST), on the other hand, aims to optimize the sparse NN connectivity jointly with model parameters. Mocanu et al. (2018) and Mostafa et al. (2019) propose replacing low magnitude parameters with random connections and report improved generalization. Dettmers et al. (2019) proposed using momentum values, whereas Evci et al. (2020) used gradient estimates directly to guide the selection of new connections, reporting results that are on par with pruning algorithms. In $\ S 4 . 2$ we study these algorithms and try to understand the role of gradient flow in their success.
+
+Random Initialization of Sparse NN In training sparse NN from scratch, the vast majority of pre-exisiting work on training sparse NN has used the common initialization methods (Glorot et al., 2010; He et al., 2015) derived for dense NNs, with only a few notable exceptions. Gale et al. (2019), Liu et al. (2019), and Ramanujan et al. (2019) scaled the variance (fan-in/fan-out) of a sparse NN layer according to the layer’s sparsity, effectively using the standard initialization for a small dense layer of equivalent number of weights as in the sparse model.
+
+# 3 Analyzing Gradient Flow in Sparse Neural Networks
+
+A significant breakthrough in training very deep NNs arose in addressing the vanishing and exploding gradient problem, both at initialization, and during training. This problem was understood by analyzing the signal propagation within a DNN, and addressed in improved initialization methods (Glorot et al., 2010; He et al., 2015; Xiao et al., 2018) alongside normalization methods, such as Batch Normalization (BatchNorm) (Ioffe et al., 2015). In our work, following Wang et al. (2020), we study these problems using the gradient flow, $\nabla L ( \theta ) ^ { T } \nabla L ( \theta )$ which is the first order approximation\* of the decrease in the loss expected after a gradient step. We observe poor gradient flow for the predominant sparse NN initialization strategy and propose a solution in $\ S 3 . 1$ . Then in $\ S 3 . 2$ and $\ S 3 . 3$ we summarize Dynamic Sparse Training (DST) methods and LT hypothesis respectively.
+
+![](images/04a07b582ebe16469a1dde6a90fd8611f3f5b7342b7981e76dcd5460a05433dd.jpg)  
+Figure 1: Glorot/He Initialization for a Sparse NN. All neurons in a dense NN layer (a) have the same fan-in, whereas in a sparse NN (b) the fan-in can differ for every neuron, potentially requiring sampling from a different distribution for every neuron. The initialization derivation/fan-out variant are explained further in Appendix A.1. (c) Std. dev. of the pre-softmax output of LeNet5 with input sampled from a normal distribution, over 5 different randomly-initialized sparse NN for a range of sparsities.
+
+# 3.1 The Initialization Problem in Sparse Networks
+
+Here we analyze the gradient flow at initialization for random sparse NNs, motivating the derivation of a more general initialization for NN with heterogeneous connectivity, such as in sparse NNs. In practice, without a method such as BatchNorm (Ioffe et al., 2015), using the correct initialization can be the difference between being able to train a DNN, or not — as observed for VGG16 in our results (§4.1, Table 1). The initializations proposed by Glorot et al. (2010) and He et al. (2015) ensure that the output distribution of every neuron in a layer is of zero-mean and unit variance, and do this by sampling a Gaussian distribution with a variance based on the number of incoming/outgoing connections for all the neurons in a dense layer, as illustrated in Fig. 1a, which is assumed to be identical for all neurons in the layer.
+
+In an unstructured sparse NN however, the number of incoming/outgoing connections is not identical for all neurons in a layer, as illustrated in Fig. 1b. In Appendix A.2 we derive the initialization for this more general case. In Appendix A.1 we explain in full the generalized Glorot et al. (2010) and He et al. (2015) initialization, in the forward, backward and average use cases. Here we will focus only on explaining the generalized He et al. (2015) initialization for forward propagation, which we used in our experiments.
+
+For every weight $w _ { i j } ^ { \left[ \ell \right] } \in W ^ { n ^ { \left[ \ell \right] } \times n ^ { \left[ \ell - 1 \right] } }$ in a layer $\ell$ with $n ^ { \left[ \ell \right] }$ neurons, and mask $[ m _ { i j } ^ { [ \ell ] } ] { = } M ^ { \ell } \in [ 0 , 1 ] ^ { n ^ { [ \ell ] } \times n ^ { [ \ell - 1 ] } }$
+
+$$
+w _ { i j } ^ { \left[ \ell \right] } \sim \mathcal { N } \Bigg ( 0 , \frac { 2 } { f a n - i n _ { i } ^ { \left[ \ell \right] } } \Bigg ) , \qquad \mathrm { w h e r e } f a n - i n _ { i } ^ { \left[ \ell \right] } = \sum _ { j = 1 } ^ { n ^ { \left[ \ell - 1 \right] } } m _ { i j } ^ { \left[ \ell \right] } ,
+$$
+
+is the number of incoming connections for neuron $i$ in layer $\ell$ .
+
+In the special case of a dense layer where $m _ { i j } ^ { [ \ell ] } = 1 , \forall i , j$ , Eq. (1) reduces to the initialization proposed by (He et al., 2015) since fan- $i n _ { i } ^ { [ \ell ] } { = } n ^ { [ \ell - 1 ] } , \forall i$ . Using the dense initialization in a sparse DNN causes signal to vanish, as empirically observed in Fig. 1c), whereas our initialization keeps the variance of the signal constant. The initialization proposed by Liu et al. (2019) is a special case of ours where it is assumed fan- $i n _ { i } ^ { [ \ell ] } \equiv f a n \ - i n ^ { [ \ell ] } , \forall i ,$ , i.e. all neurons have the same number of unmasked incoming connections in a layer. Surprisingly the initialization of Liu et al. (2019) also preserves the signal in Fig. 1c (discussed in $\ S 4 . 1 \ r _ { , }$ .
+
+# 3.2 Gradient Flow during Training and Dynamic Sparse Training
+
+While initialization is important for the first training step, the gradient flow during the early stages of training is not well addressed by initialization alone, as shown by normalization methods (Ioffe et al., 2015). Our findings show that even with BatchNorm, the gradient flow during training in unstructured sparse NNs is poor.
+
+Recently, a promising new approach to training sparse NNs has emerged — Dynamic Sparse Training (DST) — that learns connectivity adaptively during training, showing significant improvements over baseline methods that use a fixed mask. These methods perform periodic updates on the sparse connectivity of each layer: commonly replacing least magnitude connections with new connections selected using various criteria. We consider two of these methods: Sparse Evolutionary Training (SET) (Mocanu et al.,
+
+2018), which chooses new connections randomly and Rigged Lottery (RigL) (Evci et al., 2019), which chooses connections with high gradient magnitude. RigL improves over SET and matches pruning performance with sufficient training time. Since these methods have only recently been proposed, there is a lack of understanding of why and how these methods achieve better results.
+
+# 3.3 Lottery Ticket Hypothesis
+
+A recent approach for training unstructured sparse NNs while achieving similar generalization to the original dense solution is the Lottery Ticket Hypothesis (LTH) (Frankle et al., 2019a). Notably, rather than training a pruned NN structure from random initialization, the LTH uses the dense initialization from which the pruning solution was trained/derived from.
+
+Definition [Lottery Ticket Hypothesis]: Given a NN $f$ with a parameter vector $\theta$ and an optimization function $O ^ { N } ( f , \theta ) \dot { = } \theta ^ { N }$ , which gives the optimized parameters of $f$ after $N$ training steps, there exists a sparse sub-network characterized by the binary mask $M$ such that for some iteration $K$ , $\mathbf { \partial } ^ { \circ ^ { N } } ( f , \theta ^ { K } { * } M )$ performs as well as $O ^ { N } ( f , \theta ) * M$ , whereas the model trained from another random initialization $\theta _ { S }$ , using the same mask $O ^ { N } ( f , \theta _ { S } { * } \dot { M } )$ , typically does not\*. Frankle et al. (2019a) initially claimed the LTH held for $K = 0$ , but later revised this to $N { \gg } K { \ge } 0$ (Frankle et al., 2019b; Liu et al., 2019).
+
+LTs enjoy significantly faster convergence compared to regular NN training but require the connectivity mask as found by the pruning solution (Frankle et al., 2019a) along with values from early training (Frankle et al., 2019b). Given the importance of the early phase of training (Frankle et al., $2 0 2 0 \mathrm { c }$ ; Lewkowycz et al., n.d.), it is natural to ask about the difference between lottery tickets and the solution they are derived from (i.e. pruning solutions). Answering this question can help us understand if the success of LTs is primarily due to its relation to the solution, or if we can identify generalizable characteristics that help with sparse NNs training.
+
+# 4 Experiments
+
+Here we show empirically that (1) sparsity-aware initialization improves gradient flow at initialization for all methods, and achieves higher generalization for networks without BatchNorm, (2) the mask updates of DST methods increase gradient flow and create new negative eigenvalues in the Hessian; which we believe to be the main factor for improved generalization, (3) lottery tickets have poor gradient flow, however they achieve good performance by effectively re-learning the pruning solution, meaning they do not address the problem of training sparse NNs in general. Our experiments include the following settings: LeNet5 on MNIST, VGG16 on ImageNet-2012 and ResNet-50 on ImageNet-2012. Experimental details can be found in Appendix $\mathbf { B } ^ { \dagger }$ .
+
+# 4.1 Gradient Flow at Initialization
+
+In this section, we measure the gradient flow over the course of the training (Fig. 2) and evaluate the performance of our generalized He initialization method (Table 1), and that proposed by Liu et al. (2019), over the commonly used masked dense initialization. Additional gradient flow plots for the remaining methods are shared in Appendix C. Sparse NN initialized using the initialization distribution of a dense model (Scratch in Fig. 2) start in a flat region where gradient flow is very small and don’t make any early progress. Learning starts after 1000 iterations for LeNet5 and 5000 for VGG-16, however, their generalization is sub-optimal. Liu et al. (2019) claim their proposed initialization has no empirical effect as compared to the masked dense initialization‡. Although technically incorrect (see $\ S 3 . 1 \ r _ { , }$ ), our results show their method to be largely as effective as our proposed initialization. This indicates that the assumption of a mask having roughly uniform mask sparsity is sufficient for the masks we considered. Both of these initializations remedy the vanishing gradient problem at initialization (Scratch+ in Fig. 2) and result in better generalization for all methods. For instance, improved initialization results in an $11 \%$ improvement in Top-1 accuracy for VGG16 (62.52 vs 51.81). While initialization is extremely important for NNs without BatchNorm and skip connections, its effect on modern architectures, such as Resnet-50, is limited (Evci et al., 2019; Frankle et al., 2020b; Zhang et al., 2019). We confirm these observations in our ResNet-50 experiments in which, despite some initial improvement in gradient flow, our initialization seems to have no effect on final generalization. We observe significant increases in gradient norm after each learning rate drop (due to increased variance in gradients), which suggests studying gradient norm in the later part of the training might not be helpful. On the other hand, we observe a significant difference in gradient flow during training between sparse networks and small dense models of a similar parameter count. Can the performance gap between static-sparse and dense models be explained by this difference?
+
+Table 1: Results of Trained Sparse/Dense Models from Different Initializations. The initializations proposed in Eq. (1) (Ours) and Liu et al. (2019) improve generalization consistently over masked dense (Original) except for in ResNet50. Note that VGG16 trained without a sparsity-aware initialization fails to converge in some instances. Baseline corresponds to the original dense architecture, whereas Small Dense corresponds to a smaller dense model with approximately the same parameter count as the sparse models.   
+
+<table><tr><td></td><td colspan="3">MNIST</td><td colspan="6">ImageNet-2012</td></tr><tr><td></td><td colspan="3">LeNet5 (95% sparse)</td><td colspan="3">VGG16 (80% sparse)</td><td colspan="3">ResNet50 (80% sparse)</td></tr><tr><td>Baseline Lottery</td><td colspan="3">99.21±0.07</td><td colspan="3">69.25±0.13</td><td colspan="3">76.75±0.12</td></tr><tr><td>Small</td><td colspan="3">98.26±0.27 98.21±0.46</td><td colspan="3">0.10±0.01</td><td colspan="3">75.75±0.12*</td></tr><tr><td>Dense</td><td colspan="3"></td><td colspan="3">61.75±0.09</td><td colspan="3">71.95±0.24</td></tr><tr><td></td><td>Original</td><td>Liu et al.</td><td>Ours</td><td>Original</td><td>Liu et al.</td><td>Ours</td><td>Original</td><td>Liu et al.</td><td>Ours</td></tr><tr><td>Scratch</td><td>62.99±42.16</td><td>96.64±0.83</td><td>97.70±0.09</td><td>51.81±3.02</td><td>62.71±0.05</td><td>62.52±0.10</td><td>70.58±0.18</td><td>70.72±0.16</td><td>70.63±0.22</td></tr><tr><td>SET</td><td>63.33±42.44</td><td>97.77±0.31</td><td>98.16±0.06</td><td>53.55±1.03</td><td>63.19��0.26</td><td>63.13±0.15</td><td>72.93±0.27</td><td>72.77±0.27</td><td>72.56±0.14</td></tr><tr><td>RigL</td><td>80.82±34.74</td><td>98.14±0.17</td><td>98.13±0.09</td><td>37.15±26.20</td><td>63.69±0.02</td><td>63.56±0.06</td><td>74.41±0.05</td><td>74.38±0.10</td><td>74.38±0.01</td></tr></table>
+
+Used late-rewinding (i.e. $K = 5 0 0 0$ ).
+
+![](images/a073c98ef67cf22863d25c16e5f1222e9c5893298fb0819227c3604425ba519a.jpg)  
+Figure 2: Gradient Flow of Sparse Models during Training. Gradient flow during training averaged over multiple runs, $^ \bullet + ^ { \bullet }$ indicates training runs with our proposed sparse initialization and Small Dense corresponds to training of a dense network with same number of parameters as the sparse networks. Lottery ticket runs for ResNet-50 include late-rewinding.
+
+![](images/4fbe61619a3e6f65fa649e3fdaf356a57509ae34fbfe7804e6b3c6cbed329004.jpg)  
+Figure 3: Effect of Mask Updates in Dynamic Sparse Training. Effect of mask updates on the gradient norm. RigL Inverted chooses connections with least magnitude. We measure the gradient norm before and after the mask updates and plot the $\Delta$ . $^ { \bullet } + ^ { \bullet }$ indicates proposed initialization and used in MNIST experiments.
+
+![](images/6b459893b49afa81ee949f03ba927c11c928a4cc8b55eea69a242b568c28a8bd.jpg)  
+Figure 4: Lottery Tickets Are Biased Towards the Pruning Solution, Unlike Random Initialization. A cartoon illustration of the loss landscape of a sparse model, after it is pruned from a dense solution to create a LT sub-network. A lottery ticket initialization is within the basin of attraction of the pruned model’s solution. In contrast a random initialization is unlikely to be close to the dense solution’s basin.
+
+# 4.2 Gradient Flow during Training and Dynamic Sparse Training
+
+In Fig. 2 we observed improved gradient flow for RigL. In this section we focus on those iterations in which the sparse connectivity is updated, and measure the change in gradient flow along with the Hessian spectrum. We also run the inverted baseline for RigL (RigL Inverted), in which the growing criteria is reversed and connections with least gradient magnitudes are activated.
+
+DST methods such as RigL replace low saliency connections during training. Assuming the pruned connections indeed have a low impact on the loss, we might expect to see increased gradient norm after new connections are activated, especially in the case of RigL, which picks new connections with high magnitude gradients. In Fig. 3 we confirm that RigL updates increase the norm of the gradient significantly, especially in the first half of training, whereas SET, which picks new connections randomly, seems to be less effective at this. Using the inverted RigL criteria doesn’t improve the gradient flow, as expected, and without this RigL’s performance degrades ( $7 3 . 8 3 { \pm } 0 . 1 2$ for ResNet-50 and $9 2 . 7 1 { \pm } 7 . 6 7$ for LeNet5). These results suggest that improving gradient flow early in training might be the key for training sparse networks and that is what RigL appears to be doing. Additional plots for different initialization methods are shared in Appendix C.
+
+RigL falls short of matching Small-Dense performance while constantly having higher gradient flow during the training, which highlights a limitation of looking solely at gradient flow. When the gradient is zero, or uninformative due to the error term of the approximation, analyzing the Hessian could provide additional insights (Ghorbani et al., 2019; Papyan, 2019; Sagun et al., 2017). In Appendix E, we show the Hessian spectrum before and after sparse connectivity updates. After RigL updates we observe more negative eigenvalues with significantly larger magnitudes as compared to SET. On the other hand, small dense models have smaller positive outlier eigenvalues while having significantly larger negative ones; which is again a sign of better conditioned optimization. We leave investigating the relationship between gradient flow and the Hessian further as a future work.
+
+# 4.3 Why Lottery Tickets are Successful
+
+We found that LTs do not improve gradient flow, either at initialization, or early in training, as shown in Fig. 2. This may be surprising given the apparent success of LTs, however the questions posed in $\ S 3 . 3$ present an alternative hypothesis for the ease of training from a LT initialization. Here we present results showing that indeed (1) LTs initializations are consistently closer to the pruning solution than a random initialization, (2) trained LTs (i.e. LT solutions) consistently end up in the same basin as the pruning solution and (3), LT solutions are highly similar to pruning solutions under various function similarity measures. Our resulting understanding of LTs in the context of the pruning solution and the loss landscape is illustrated in Fig. 4.
+
+Experimental Setup To investigate the relationship between the pruned and LT solutions we perform experiments on two models/datasets: a $9 5 \%$ sparse $\mathrm { L e N e t } 5 ^ { \ S }$ architecture (LeCun et al., 1989) trained on MNIST (where the original LT formulation works, i.e. $K { = } 0$ ), and an $80 \%$ sparse ResNet-50 (Wu et al., 2018) on ImageNet-2012 (Russakovsky et al., 2015) (where $K = 0$ doesn’t work (Frankle et al., 2019b)), for which we use values from $K = 2 0 0 0$ ${ \approx } 6 ^ { \mathrm { t h } }$ epoch). In both cases, we find a LT initialization by pruning each layer of a dense NN separately using magnitude-based iterative pruning (Zhu et al., 2018). Further details about our experiments can be found in Appendix B.
+
+Lottery Tickets Are Close to the Pruning Solution We train 5 different models using different seeds from both scratch (random) and LT initializations, the results of which are in Figs. 5b and 5e. These networks share the same pruning mask and therefore lie in the same solution space. We visualize distances between initial and final points of these experiments in Figs. 5a and 5d using 2D Multi-dimensional Scaling (MDS) (Kruskal, 1964) embeddings. LeNet5/MNIST: In Fig. 5b, we provide the average L2 distance to the pruning solution at initialization $( d _ { i n i t } )$ , and after training $( d _ { f i n a l } )$ . We observe that LT initializations start significantly closer to the pruning solution on average $( d _ { i n i t } = 1 3 . 6 1$ v.s. 17.46). After training, LTs end up more than $3 \times$ closer to the pruning solution compared to scratch. Resnet-50/ImageNet-2012: We observe similar results for Resnet-50/ImageNet-2012. LTs, again, start closer to the pruning solution, and solutions are $5 \times$ closer $( d _ { f i n a l } = 3 9 . 3 5 $ v.s. 215.98). With these observations, non-random initial loss values for LT initialization reported first by (Zhou et al., 2019) seem reasonable. LTs are biased towards the pruning solution they are derived from, but are they in the same basin?
+
+![](images/9e2b3d45329c14d1d562527240cd85dffbee36fbfe6beba82b589dfa1412c87b.jpg)  
+Figure 5: MDS Embeddings/L2 Distances: (a, d): 2D Multi-dimensional Scaling (MDS) embedding of sparse NNs with the same connectivity/mask; (b, e): the average L2-distance between a pruning solution and other derived sparse networks; (c, f): linear path between the pruning solution $\alpha { = } 1 . 0$ ) and LT/scratch at both initialization, and solution (end of training). Top and bottom rows are for MNIST/LeNet5 and ImageNet-2012/ResNet-50 respectively.
+
+Lottery Tickets are in the Pruning Solution Basin Investigating paths between different solutions is a popular tool for understanding how various points in parameter space relate to each other in the loss landscape (Draxler et al., 2018; Evci et al., 2019; Fort et al., 2020; Frankle et al., 2020a; Garipov et al., 2018; Goodfellow et al., 2015). For example, Frankle et al. (2019b) use linear interpolations to show that LTs always go to the same basin¶ when trained in different data orders. In Figs. 5c and 5f we look at the linear paths between pruning solution and 4 other points: LT initialization/solution and random (scratch) initialization/solution. Each experiment is repeated 5 times with different random seeds, and mean values are provided with $80 \%$ confidence intervals. In both experiments we observe that the linear path between LT initialization and the pruning solution decreases faster compared to the path that originates from scratch initialization. After training, the linear paths towards the pruning solution change drastically. The path from the scratch solution depicts a loss barrier; the scratch solution seems to be in a different basin than the pruning solution||. In contrast, LTs are linearly connected to the pruning solution in both small and large-scale experiments indicating that LTs have the same basin of attraction as the pruning solutions they are derived from. While it seems likely, these results do not however explicitly show that the LT and pruning solutions have learned similar functions.
+
+Table 2: Ensemble & Prediction Disagreement. We compare the function similarity (Fort et al., 2020) with the original pruning solution and ensemble generalization over 5 sparse models, trained from random initializations and LTs. As a baseline, we also show results for 5 pruned models trained from different random initializations. See Appendix F for the complete results.   
+
+<table><tr><td>LT</td><td>Initialization</td><td>(Top-1) Test Acc.</td><td>Ensemble</td><td>Disagree.</td><td>Disagree. w/ Pruned</td></tr><tr><td rowspan="6">LeNet5 MNIST</td><td></td><td>98.52±0.02</td><td>98.58</td><td>0.0043±0.0006</td><td>0.0089±0.0002</td></tr><tr><td>Scratch</td><td>97.04±0.15</td><td>98.00</td><td>0.0316±0.0023</td><td>0.0278±0.0020</td></tr><tr><td>Scratch (Diff. Init.)</td><td>97.19±0.33</td><td>98.43</td><td>0.0352±0.0037</td><td>0.0278±0.0032</td></tr><tr><td>Prune Restart</td><td>98.60±0.01</td><td>98.63</td><td>0.0027±0.0003</td><td>0.0077±0.0003</td></tr><tr><td>Pruned Soln.</td><td>98.53</td><td>1</td><td></td><td></td></tr><tr><td>5 Diff. Pruned</td><td>98.30±0.23</td><td>99.07</td><td>0.0214±0.0023</td><td>0.0197��0.0019*</td></tr><tr><td rowspan="5">ResNet50 ImageNet</td><td>LT</td><td>75.73±0.08</td><td>76.27</td><td>0.0894±0.0009</td><td>0.0941±0.0009</td></tr><tr><td>Scratch</td><td>71.16±0.13</td><td>74.05</td><td>0.2039±0.0013</td><td>0.2033±0.0012</td></tr><tr><td>Pruned Soln.</td><td>75.60</td><td></td><td></td><td></td></tr><tr><td>5 Diff. Pruned</td><td></td><td>1</td><td></td><td></td></tr><tr><td></td><td>75.65±0.13</td><td>77.80</td><td>0.1620±0.0008</td><td>0.1623±0.0011*</td></tr></table>
+
+\* Here we compare 4 different pruned models with the pruning solution LT/Scratch are derived from.
+
+Lottery Tickets Learn Similar Functions to the Pruning Solution Fort et al. (2020) motivate deep ensembles by empirically showing that models starting from different random initializations typically learn different solutions, as compared to models trained from similar initializations. Here we adopt the analysis of (Fort et al., 2020), but in comparing LT initializations and random initializations using fractional disagreement. The fractional disagreement with the pruning solution is the fraction of class predictions over which the LT and scratch models disagree with the pruning solution they were derived from. In Table 2 we show the mean fractional disagreement over all pairs of models. We run two versions of scratch training: (1) Scratch (Diff. Init. different weight initialization and different data order (2) Scratch same weight initialization and different data order for 5 different seeds the experiments are ran. Finally, we restart training starting from the pruning solution (Prune Restart) using, again, 5 different data orders.
+
+The results presented in Table 2 suggest that all 5 LTs models converge on a solution almost identical to the pruning solution. Interestingly, the 5 LT models are even more similar to each other (Disagree. column) than the pruning solution, possibly because they share an initialization and training is stable (Frankle et al., 2019b). The disagreement of Prune Restart solutions with the original pruning solution matches the disagreement of lottery solutions; showing the extent of similarity between LT and pruning solutions.
+
+Our results show that having a fixed initialization alone can not explain the low disagreement observed for LT experiments as Scratch solutions obtain an average disagreement of 0.0316 despite using the same initialization, which is almost 10 times more than the LT solutions (0.0043). Finding different LT initialization is costly, however using a different initialization in Scratch (Diff. Init.) training is free as the initializations are random. Using different initializations we can obtain more diverse solutions and thus achieve higher ensemble accuracy. As suggested by the analysis of Fort et al. (2020), ensembles of different solutions are more robust, and generalize better, than ensembles of similar solutions. An ensemble of $5 \mathrm { L T }$ models with low disagreement doesn’t significantly improve generalization as compared to an ensemble of 5 different pruning solutions with similar individual test accuracy. We further demonstrate these results by comparing the output probability distributions using the Kullback–Leibler Divergence (KL), and Jensen–Shannon Divergence (JSD) in Appendix F.
+
+Implications: (a) Rewinding of LTs. Frankle et al. (2019b, 2020a) argued that LTs work when the training is stable, and thus converges to the same basin when trained with different data sampling orders. In $\ S 4 . 3$ , we show that this basin is the same one found by pruning, and since the training converges to the same basin as before, we expect to see limited gains from rewinding if any. This is partially confirmed by Renda et al. (2020) which shows that restarting the learning rate schedule from the pruning solution performs better than rewinding the weights. (b) Transfer of LTs. Given the close relationship between LTs and pruning solutions, the observation that LTs trained on large datasets transfer to smaller ones, but not vice versa (Morcos et al., 2019; Sabatelli et al., 2020) can be explained by a common observation in transfer learning: networks trained in large datasets transfer to smaller ones. (c) LT’s Robustness to Perturbations. Frankle et al. (2020c) and Zhou et al. (2019) found that certain perturbations, like only using the signs of weights at initialization, do not impact LT generalization, while others, like shuffling the weights, do. Our results bring further insights to these observations: As long as the perturbation is small enough such that a LT stays in the same basin of attraction, results will be as good as the pruning solution. (d) Success of LTs. While it is exciting to see widespread applicability of LTs in different domains (Brix et al., 2020; Li et al., 2020; Venkatesh et al., 2020), the results presented in this paper suggest this success may be due to the underlying pruning algorithm (and transfer learning) rather than LT initializations themselves.
+
+# 5 Conclusion
+
+We attempted to answer the questions of (1) why training unstructured sparse networks from random initialization performs poorly and; (2) what makes Lottery Tickets (LTs) and Dynamic Sparse Training (DST) the exceptions? We identified that randomly initialized unstructured sparse Neural Networks (NNs) exhibit poor gradient flow when initialized naively and proposed an alternative initialization that scales the initial variance for each neuron separately. Furthermore we showed that modern sparse NN architectures are more sensitive to poor gradient flow during early training rather than initialization alone. We observed that this is somewhat addressed by state-of-the-art DST methods, such as Rigged Lottery (RigL), which significantly improves gradient flow during early training over traditional sparse training methods. Finally, we show that LTs do not improve gradient flow at either initialization or during training, but rather their success lies in effectively re-learning the original pruning solution they are derived from. We showed that a LTs initialization resides within the same basin of attraction as the pruning solution and, furthermore, when trained the LT solution learns a highly similar solution to the pruning solution. These findings suggest that LTs are fundamentally limited in their potential for improving the training of sparse NNs more generally.
+
+# References
+
+Brix, Christopher, Parnia Bahar, and Hermann Ney (2020). “Successfully Applying the Stabilized Lottery Ticket Hypothesis to the Transformer Architecture”. In: ArXiv. URL: https://arxiv.org/abs/ 2005.03454.   
+Castellano, Giovanna, Anna Maria Fanelli, and Marcello Pelillo (1997). “An iterative pruning algorithm for feedforward neural networks”. In: IEEE Transactions on Neural Networks. ISSN: 1045-9227. DOI: 10.1109/72.572092.   
+Dettmers, Tim and Luke Zettlemoyer (2019). “Sparse Networks from Scratch: Faster Training without Losing Performance”. In: ArXiv. URL: http://arxiv.org/abs/1907.04840.   
+Draxler, Felix, Kambis Veschgini, Manfred Salmhofer, and Fred A Hamprecht (2018). “Essentially No Barriers in Neural Network Energy Landscape”. In: International Conference on Machine Learning.   
+Evci, Utku, Trevor Gale, Jacob Menick, Pablo Samuel Castro, and Erich Elsen (2020). “Rigging the Lottery: Making All Tickets Winners”. In: Proceedings of Machine Learning and Systems 2020.   
+Evci, Utku, Fabian Pedregosa, Aidan N. Gomez, and Erich Elsen (2019). “The Difficulty of Training Sparse Neural Networks”. In: ArXiv. URL: http://arxiv.org/abs/1906.10732.   
+Fort, Stanislav, Huiyi Hu, and Balaji Lakshminarayanan (2020). “Deep Ensembles: A Loss Landscape Perspective”. In: International Conference on Learning Representations.   
+Frankle, Jonathan and Michael Carbin (2019a). “The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks”. In: 7th International Conference on Learning Representations (ICLR).   
+Frankle, Jonathan, Gintare Karolina Dziugaite, Daniel M. Roy, and Michael Carbin (2019b). “Stabilizing the Lottery Ticket Hypothesis”. In: ArXiv. URL: https://arxiv.org/abs/1903.01611. (2020a). “Linear Mode Connectivity and the Lottery Ticket Hypothesis”. In: Proceedings of the International Conference on Machine Learning. (2020b). “Pruning Neural Networks at Initialization: Why are We Missing the Mark?” In: ArXiv. URL: https://arxiv.org/abs/2009.08576.   
+Frankle, Jonathan, David J. Schwab, and Ari S. Morcos (2020c). “The Early Phase of Neural Network Training”. In: International Conference on Learning Representations.   
+Gale, Trevor, Erich Elsen, and Sara Hooker (2019). “The State of Sparsity in Deep Neural Networks”. In: ArXiv. URL: http://arxiv.org/abs/1902.09574.   
+Garipov, Timur, Pavel Izmailov, Dmitrii Podoprikhin, Dmitry P Vetrov, and Andrew Gordon Wilson (2018). “Loss Surfaces, M Connectivity, and Fast Ensembling of DNNs”. In: Advances in Neural Information Processing Systems.   
+Ghorbani, Behrooz, Shankar Krishnan, and Ying Xiao (2019). “An Investigation into Neural Net Optimization via Hessian Eigenvalue Density”. In: Proceedings of the 36th International Conference on Machine Learning. URL: http://proceedings.mlr.press/v97/ghorbani19b.html.   
+Glorot, Xavier and Yoshua Bengio (2010). “Understanding the difficulty of training deep feedforward neural networks”. In: Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (AISTATS).   
+Goodfellow, Ian J., Oriol Vinyals, and Andrew M. Saxe (2015). “Qualitatively characterizing neural network optimization problems”. In: International Conference on Learning Representations.   
+Han, Song, Xingyu Liu, Huizi Mao, Jing Pu, Ardavan Pedram, Mark A. Horowitz, and William J. Dally (2016). “EIE: Efficient Inference Engine on Compressed Deep Neural Network”. In: Proceedings of the 43rd International Symposium on Computer Architecture.   
+Han, Song, Jeff Pool, John Tran, and William Dally (2015). “Learning both weights and connections for efficient neural network”. In: Advances in neural information processing systems.   
+Hanson, Stephen Jose and Lorien Y. Pratt (1988). “Comparing biases for minimal network construction ´ with back-propagation”. In: Proceedings of the 1st International Conference on Neural Information Processing Systems (NIPS). Morgan Kaufmann.   
+Hassibi, B. and D. Stork (1993). “Second order derivatives for network pruning: Optimal Brain Surgeon”. In: Advances in Neural Information Processing Systems.   
+He, Kaiming, Xiangyu Zhang, Shaoqing Ren, and Jian Sun (2015). “Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification”. In: IEEE International Conference on Computer Vision (ICCV).   
+Ioffe, Sergey and Christian Szegedy (2015). “Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift”. In: Proceedings of the 32Nd International Conference on International Conference on Machine Learning.   
+Kruskal, J. (1964). “Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis”. In: Psychometrika. DOI: 10.1007/BF02289565.   
+Kusupati, Aditya, Vivek Ramanujan, Ragha Somani, Mitchell Wortsman, Prateek Jain, Sham Kakade, and Ali Farhadi (2020). “Soft Threshold Weight Reparameterization for Learnable Sparsity”. In: Proceedings of the International Conference on Machine Learning.   
+LeCun, Yann, Bernhard Boser, John S. Denker, Donnie Henderson, Richard E. Howard, Wayne Hubbard, and Lawrence D. Jackel (1989). “Backpropagation applied to handwritten zip code recognition”. In: Neural Computation. ISSN: 0899-7667. DOI: 10.1162/neco.1989.1.4.541.   
+LeCun, Yann, John S. Denker, and Sara A. Solla (1990). “Optimal Brain Damage”. In: Advances in Neural Information Processing Systems.   
+Lee, Namhoon, Thalaiyasingam Ajanthan, and Philip H. S. Torr (2019). “SNIP: Single-shot Network Pruning based on Connection Sensitivity”. In: International Conference on Learning Representations (ICLR), 2019.   
+Lewkowycz, Aitor, Yasaman Bahri, Ethan Dyer, Jascha Sohl-Dickstein, and Guy Gur-Ari (n.d.). “The large learning rate phase of deep learning: the catapult mechanism”. In: Arxiv (). URL: https : //arxiv.org/pdf/2003.02218.   
+Li, Bai, Shiqi Wang, Yunhan Jia, Yantao Lu, Zhenyu Zhong, Lawrence Carin, and Suman Jana (2020). “Towards Practical Lottery Ticket Hypothesis for Adversarial Training”. In: ArXiv. URL: https: //arxiv.org/abs/2003.05733.   
+Liu, Zhuang, Mingjie Sun, Tinghui Zhou, Gao Huang, and Trevor Darrell (2019). “Rethinking the Value of Network Pruning”. In: International Conference on Learning Representations.   
+Louizos, Christos, Karen Ullrich, and Max Welling (2017). “Bayesian compression for deep learning”. In: Advances in Neural Information Processing Systems.   
+Malach, Eran, Gilad Yehudai, Shai Shalev-Shwartz, and Ohad Shamir (2020). “Proving the Lottery Ticket Hypothesis: Pruning is All You Need”. In: ArXiv. URL: http://arxiv.org/abs/2002.00585.   
+Mocanu, Decebal Constantin, Elena Mocanu, Peter Stone, Phuong H Nguyen, Madeleine Gibescu, and Antonio Liotta (2018). “Scalable training of artificial neural networks with adaptive sparse connectivity inspired by network science”. In: Nature Communications.   
+Molchanov, Dmitry, Arsenii Ashukha, and Dmitry Vetrov (2017). “Variational Dropout Sparsifies Deep Neural Networks”. In: Proceedings of the 34th International Conference on Machine Learning. Proceedings of Machine Learning Research. International Convention Centre, Sydney, Australia: PMLR. URL: http://proceedings.mlr.press/v70/molchanov17a.html.   
+Molchanov, Pavlo, Stephen Tyree, Tero Karras, Timo Aila, and Jan Kautz (2016). “Pruning Convolutional Neural Networks for Resource Efficient Inference”. In: ArXiv. URL: http://arxiv.org/abs/ 1611.06440.   
+Morcos, Ari, Haonan Yu, Michela Paganini, and Yuandong Tian (2019). “One ticket to win them all: generalizing lottery ticket initializations across datasets and optimizers”. In: Advances in Neural Information Processing Systems 32.   
+Mostafa, Hesham and Xin Wang (2019). “Parameter efficient training of deep convolutional neural networks by dynamic sparse reparameterization”. In: Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA. URL: http: //proceedings.mlr.press/v97/mostafa19a.html.   
+Mozer, Michael C. and Paul Smolensky (1989a). “Using Relevance to Reduce Network Size Automatically”. In: Connection Science. DOI: 10.1080/09540098908915626.   
+– (1989b). “Skeletonization: A Technique for Trimming the Fat from a Network via Relevance Assessment”. In: Advances in Neural Information Processing Systems 1.   
+Papyan, Vardan (2019). “Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians”. In: ArXiv. URL: https://arxiv.org/abs/1901.08244.   
+Ramanujan, Vivek, Mitchell Wortsman, Aniruddha Kembhavi, Ali Farhadi, and Mohammad Rastegari (2019). “What’s Hidden in a Randomly Weighted Neural Network?” In: ArXiv. URL: http://arxiv. org/abs/1911.13299.   
+Renda, Alex, Jonathan Frankle, and Michael Carbin (2020). “Comparing Rewinding and Fine-tuning in Neural Network Pruning”. In: International Conference on Learning Representations. URL: https: //openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } }$ S1gSj0NKvB.   
+Russakovsky, Olga, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei (2015). “ImageNet Large Scale Visual Recognition Challenge”. In: International Journal of Computer Vision (IJCV).   
+Sabatelli, Matthia, Mike Kestemont, and Pierre Geurts (2020). “On the Transferability of Winning Tickets in Non-Natural Image Datasets”. In: ArXiv. URL: https://arxiv.org/abs/2005.05232.   
+Sagun, Levent, Utku Evci, V. Ugur Guney, Yann Dauphin, and L ¨ eon Bottou (2017). “Empirical Analysis ´ of the Hessian of Over-Parametrized Neural Networks”. In: International Conference on Learning Representations.   
+Setiono, Rudy (1997). “A Penalty-Function Approach for Pruning Feedforward Neural Networks”. In: Neural Computation. DOI: 10.1162/neco.1997.9.1.185.   
+Sietsma, Jocelyn and Robert J.F. Dow (1988). “Neural net pruning-why and how”. In: IEEE International Conference on Neural Networks. DOI: 10.1109/ICNN.1988.23864.   
+Tanaka, Hidenori, Daniel Kunin, Daniel L. K. Yamins, and Surya Ganguli (2020). “Pruning neural networks without any data by iteratively conserving synaptic flow”. In: ArXiv. URL: https://arxiv.org/ abs/2006.05467.   
+van der Walt, S., S. C. Colbert, and G. Varoquaux (2011). “The NumPy Array: A Structure for Efficient Numerical Computation”. In: Computing in Science Engineering.   
+Venkatesh, Bindya, Jayaraman J. Thiagarajan, Kowshik Thopalli, and Prasanna Sattigeri (2020). “Calibrate and Prune: Improving Reliability of Lottery Tickets Through Prediction Calibration”. In: ArXiv. URL: http://arxiv.org/abs/2002.03875.   
+Wang, Chaoqi, Guodong Zhang, and Roger Grosse (2020). “Picking Winning Tickets Before Training by Preserving Gradient Flow”. In: International Conference on Learning Representations. URL: https: //openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } }$ SkgsACVKPH.   
+Wortsman, Mitchell, Ali Farhadi, and Mohammad Rastegari (2019). “Discovering Neural Wirings”. In: Advances in Neural Information Processing Systems.   
+Wu, Songtao, Shenghua Zhong, and Yan Liu (2018). “Deep residual learning for image steganalysis”. In: Multimedia Tools and Applications.   
+Xiao, Lechao, Yasaman Bahri, Jascha Sohl-Dickstein, Samuel S Schoenholz, and Jeffrey Pennington (2018). “Dynamical isometry and a mean field theory of cnns: How to train 10,000-layer vanilla convolutional neural networks”. In: International Conference on Machine Learning (ICML).   
+Zhang, Hongyi, Yann N. Dauphin, and Tengyu Ma (2019). “Fixup Initialization: Residual Learning Without Normalization”. In: International Conference on Learning Representations.   
+Zhou, Hattie, Janice Lan, Rosanne Liu, and Jason Yosinski (2019). “Deconstructing lottery tickets: Zeros, signs, and the supermask”. In: Advances in Neural Information Processing Systems.   
+Zhu, Michael and Suyog Gupta (2018). “To Prune, or Not to Prune: Exploring the Efficacy of Pruning for Model Compression”. In: International Conference on Learning Representations Workshop.
+
+![](images/5773d78e5b86fc9df9bd70bd63e22146d6f77603a20b6eba653091a96cc49890.jpg)  
+Figure 6: Glorot/He Initialization for a Sparse NN. (Glorot et al., 2010; He et al., 2015) restrict the outputs of all neurons to be zero-mean and of unit variance. All neurons in a dense NN layer (a) have the same fan-in/fan-out, whereas in a sparse NN (b) the fan-in/fan-out can differ for every neuron, potentially requiring sampling from a different distribution for every neuron. The fan-in matrix contains the values used in Eq. (1) for each neuron.
+
+# A Glorot/He Initialization Generalized to Neural Networks with Heterogeneous Connectivity: Full Explanation/Derivation
+
+Here we derive the full generalized initialization for both the forwards/backwards cases (i.e. fan-in/fan-out), refer to Fig. 6 for an illustration of how the connectivity for the fan-in/fan-out cases are determined for each neuron.
+
+# A.1 Generalized Glorot/He Initialization: Backwards, Forwards and Average Cases
+
+For every weight $w _ { i j } ^ { \left[ \ell \right] } \in W ^ { n ^ { \left[ \ell \right] } \times n ^ { \left[ \ell - 1 \right] } }$ in a layer $\ell$ with $n ^ { \left[ \ell \right] }$ neurons, connecting neuron $i$ in layer $\ell$ to neuron j in layer (\`−1) with n[\`−1] neurons, and weight mask [m[\`]ij ] = M \` ∈ [0,1]n[\`]×n[\`−1],
+
+$$
+\begin{array}{c} \begin{array} { r l } & { \mathrm { G l o r o t ~ e t ~ a l . ~ } ( 2 0 1 0 ) : \ w _ { i j } ^ { [ \ell ] } \sim \mathcal { N } \big ( 0 , \frac { 1 } { u } \big ) } \\ & { \mathrm { H e ~ e t ~ a l . ~ } ( 2 0 1 5 ) : \quad w _ { i j } ^ { [ \ell ] } \sim \mathcal { N } \big ( 0 , \frac { 2 } { u } \big ) } \end{array} \mathrm { w h e r e ~ } u = \left\{ \begin{array} { l l } { f a n \cdot i n _ { i } ^ { [ \ell ] } } & { ( \mathrm { f o r w a r d } ) } \\ { f a n - o u l _ { j } ^ { [ \ell ] } } & { ( \mathrm { b a c k w a r d } ) } \\ { \left( f a n \cdot i n _ { i } ^ { [ \ell ] } + f a n - o u l _ { j } ^ { [ \ell ] } \right) / 2 } & { ( \mathrm { a v e r a g e } ) } \end{array} \right.  \end{array}
+$$
+
+where,
+
+$$
+f a n { - } i n _ { i } ^ { [ \ell ] } { = } \sum _ { j = 1 } ^ { n ^ { [ \ell - 1 ] } } m _ { i j } ^ { [ \ell ] } , \qquad f a n { - } o u t _ { j } ^ { [ \ell ] } { = } \sum _ { i = 1 } ^ { n ^ { [ \ell ] } } m _ { i j } ^ { [ \ell ] } ,
+$$
+
+are the number of incoming and outgoing connections respectively. In the special case of a dense layer where $m _ { i j } ^ { [ \ell ] } = 1 , \forall i , j$ , Eq. (1) reduces to the initializations proposed by (Glorot et al., 2010; He et al., 2015) since fan- $i n _ { i } ^ { [ \ell ] } = n ^ { [ \ell - 1 ] } , \forall i$ , and fan- ${ \cdot o u t _ { j } ^ { [ \ell ] } } = n ^ { [ \ell ] } , \forall j$ .
+
+# A.2 Derivation: Fixed Mask, Forward Propagation
+
+Given a sparse NN, where the output of a neuron $a _ { i }$ is given by, $\begin{array} { r } { a _ { i } ^ { [ \ell ] } ~ = ~ f ~ \left( z _ { i } ^ { [ \ell ] } \right) } \end{array}$ , where $\begin{array} { r } { z _ { i } ^ { [ \ell ] } = \sum _ { j } ^ { n ^ { [ \ell - 1 ] } } m _ { i j } ^ { [ \ell ] } w _ { i j } ^ { [ \ell ] } a _ { j } ^ { [ \ell - 1 ] } } \end{array}$ = Pn[\`−1]j m[i where the ou $m _ { i j } ^ { [ \ell ] } \in M ^ { [ \ell ] }$ and eviou $w _ { i j } ^ { \left[ \ell \right] } \in W ^ { \left[ \ell \right] }$ are the mame the mask - $\ell$ $a _ { j } ^ { [ \ell - 1 ] }$ $M ^ { [ \ell ] } \in \mathbb { 1 } ^ { n ^ { [ \ell ] } \times n ^ { [ \ell - 1 ] } }$ is constant, where 1n[\`]×n[\`−1] is an indicator matrix.
+
+As in Glorot et al. (2010) we want to ensure $\mathrm { V a r } ( a _ { i } ^ { [ \ell ] } ) { = } \mathrm { V a r } ( a _ { i } ^ { [ \ell - 1 ] } )$ , and ${ \mathrm { m e a n } } ( a _ { i } ^ { [ \ell ] } ) = 0$ . Assume that $f ( x ) { \approx } x$ for $x$ close to 0, e.g. in the case of $f ( x ) = \mathrm { t a n h } ( x )$ , and that $w _ { i j } ^ { \left[ \ell \right] }$ and $a _ { j } ^ { [ \ell - 1 ] }$ are independent,
+
+$$
+\begin{array} { r l r } {  { \operatorname { V a r } ( a _ { \varepsilon } ^ { ( t - 1 ) } ) _ { \varepsilon \in \mathcal { N } _ { \varepsilon } } ( z _ { \varepsilon } ^ { ( t ) } ) } } \\ & { = \operatorname { V a r } ( ( \sum _ { j = 1 } ^ { \infty } \operatorname { V a r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } a _ { j } ^ { ( t ) } a _ { j } ^ { ( t - 1 ) } )  } \\ & { } & \\ & { } & {  = \operatorname { V a r } ( \sum _ { j = 1 } ^ { \infty } \operatorname { V a r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } a _ { j } ^ { ( t ) } a _ { j } ^ { ( t - 1 ) } ) } \\ & { = \sum _ { j = 1 } ^ { \infty } \operatorname { V a r } ( \operatorname* { V a r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } a _ { j } ^ { ( t ) } a _ { j } ^ { ( t - 1 ) } ) } \\ & { = \frac { \operatorname { V a r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } } { \beta _ { j = 1 } } ( \operatorname* { W a r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } a _ { j } ^ { ( t ) } a _ { j } ^ { ( t - 1 ) } ) } \\ & { = \frac { \operatorname { V a r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } } { \beta _ { j = 1 } } ( \operatorname* { W a r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } ) ^ { 2 } \operatorname { V a r } ( \operatorname* { W } _ { \varepsilon } ^ { ( t ) } a _ { j } ^ { ( t - 1 ) } ) } & { \quad \cdot \cdot \operatorname* { W i r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } \operatorname { V a r } ( \operatorname* { W a r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } ) - \operatorname* { W i r } _ { j \in \mathcal { N } _ { \varepsilon } } ( \mathcal { K } ) } \\ & { = \frac { \operatorname { V a r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } } { \beta _ { j = 1 } } \operatorname { V a r } ( \operatorname* { W a r } _ { j \in \mathcal { N } _ { \varepsilon } } ^ { ( t ) } a _ { j } ^ { ( t ) } a _ { j } ^ { ( t - 1 ) } ) }  \end{array}
+$$
+
+Assume $\mathrm { V a r } ( w _ { i m } ^ { [ \ell ] } ) = \mathrm { V a r } ( w _ { i n } ^ { [ \ell ] } ) , \forall n , m$ , i.e. the variance of all weights for a given neuron are the same, and $\operatorname { V a r } ( a _ { n } ^ { [ \ell - 1 ] } ) { = } \operatorname { V a r } ( a _ { m } ^ { [ \ell - 1 ] } )$ , i.e. the variance of any of the outputs of the previous layer are the same. Therefore we can simplify Eq. (8),
+
+$$
+\begin{array} { r } { \mathrm { V a r } ( a _ { i } ^ { [ \ell - 1 ] } ) = \displaystyle \sum _ { j = 1 } ^ { n ^ { [ \ell - 1 ] } } m _ { i j } ^ { [ \ell ] } \mathrm { V a r } ( w _ { i j } ^ { [ \ell ] } ) \mathrm { V a r } ( a _ { j } ^ { [ \ell - 1 ] } ) } \\ { = \mathrm { V a r } ( w _ { i j } ^ { [ \ell ] } ) \mathrm { V a r } ( a _ { j } ^ { [ \ell - 1 ] } ) \displaystyle \sum _ { j = 1 } ^ { n ^ { [ \ell - 1 ] } } m _ { i j } ^ { [ \ell ] } . } \end{array}
+$$
+
+$i$ $i n _ { i } ^ { \left[ \ell \right] }$ $\begin{array} { r } { i n _ { i } ^ { [ \ell ] } { = } \sum _ { j = 1 } ^ { n ^ { [ \ell - 1 ] } } m _ { i j } ^ { [ \ell ] } } \end{array}$ , then
+
+$$
+\begin{array} { r l r } { \mathrm { V a r } ( a _ { i } ^ { [ \ell - 1 ] } ) { = } f a n { - } i n _ { i } ^ { [ \ell ] } \mathrm { V a r } ( w _ { i j } ^ { [ \ell ] } ) \mathrm { V a r } ( a _ { j } ^ { [ \ell - 1 ] } ) } & { { } } & { } \\ { \mathrm { R e c a l l , V a r } ( a _ { i } ^ { [ \ell - 1 ] } ) { = } \mathrm { V a r } ( a _ { j } ^ { [ \ell - 1 ] } ) } & { { } } & { } \\ { { \Rightarrow } \mathrm { V a r } ( w _ { i j } ^ { [ \ell ] } ) { = } \displaystyle \frac { 1 } { f a n { - } i n _ { i } ^ { [ \ell ] } } . } & { { } } & { } \end{array}
+$$
+
+Therefore, in order to have the output of each neuron $a _ { i } ^ { [ \ell ] }$ in layer $\ell$ to have unit variance, and mean 0, we need to sample the weights for each neuron from the normal distribution,
+
+$$
+[ w _ { i j } ^ { [ \ell ] } ] \sim { \cal N } \left( 0 , \frac { 1 } { f a n { - } i n _ { i } ^ { [ \ell ] } } \right) ,
+$$
+
+where $s _ { i } ^ { \ell }$ is the sparsity of weights of the neuron with output $a _ { i }$ . For the ReLU activation function, following the derivation in He et al. (2015),
+
+$$
+[ w _ { i j } ^ { [ \ell ] } ] \sim { \cal N } \left( 0 , \frac { 2 } { f a n { - } i n _ { i } ^ { [ \ell ] } } \right) .
+$$
+
+# A.3 Fixed Mask: Backward Pass
+
+Given a sparse NN, where the output of a neuron $a _ { i }$ is given by, $a _ { i } ^ { [ \ell ] } ~ = ~ f ~ \left( z _ { i } ^ { [ \ell ] } \right)$ , where $\begin{array} { r } { z _ { i } ^ { [ \ell ] } = \sum _ { j } ^ { n ^ { [ \ell - 1 ] } } m _ { i j } ^ { [ \ell ] } w _ { i j } ^ { [ \ell ] } a _ { j } ^ { [ \ell - 1 ] } } \end{array}$ , where $m _ { i j } ^ { [ \ell ] } \in M ^ { [ \ell ] }$ and $w _ { i j } ^ { \left[ \ell \right] } \in W ^ { \left[ \ell \right] }$ are the mask and weights respectively for layer \`, and a[\`−1]j the output of the previous layer. Assume the mask $M ^ { [ \ell ] } \in \mathbb { I } ^ { n ^ { [ \ell ] } \times n ^ { [ \ell - 1 ] } }$ is constant, where 1n[\`]×n[\`−1] is an indicator matrix, and let $L \left( \theta = \{ W ^ { [ \ell ] } , \ell = 0 . . . N \} \right)$ be the loss we are optimizing. As in Glorot et al. (2010), from the backward-propagation standpoint, we want to ensure $\begin{array} { r } { \mathrm { V a r } ( { \frac { \partial L } { \partial z _ { i } ^ { [ \ell ] } } } ) { = } \mathrm { V a r } ( { \frac { \partial L } { \partial z _ { i } ^ { [ \ell - 1 ] } } } ) ) } \end{array}$ r( ∂L∂z[\`−1] )), and mean( ∂ $\mathrm { m e a n } ( \frac { \partial L } { \partial z _ { i } ^ { [ \ell ] } } ) = 0$ . Assume that $f ^ { \prime } ( 0 ) { = } 1$ ,
+
+$$
+\begin{array} { r l r } { \nabla _ { \mathbf x \in \mathcal { Q } } \frac { \partial L } { \partial \mathbf x } \Biggr ( \frac { \partial L } { \partial \mathbf x } \Biggr ) \times \nabla _ { \mathbf x \in \mathcal { W } _ { \varepsilon } ^ { 0 } } \Biggr \} } & { } & \\ & { = \nabla _ { \mathbf x } \Biggl ( \int _ { \mathbf x } ^ { \mathcal { W } _ { \varepsilon } ^ { 0 } } \nabla _ { \mathbf x } \frac { \partial L } { \partial \mathbf x ^ { \varepsilon } } \frac { \partial L } { \partial \mathbf x ^ { \varepsilon } } \Biggr ) } & \\ & { = \frac { \partial ^ { \mathcal { W } _ { \varepsilon } ^ { 0 } } } { \partial \mathbf x } \Biggr ( \mathrm { r a n } _ { \mathcal { W } _ { \varepsilon } ^ { 0 } } ^ { \mathcal { W } _ { \varepsilon } ^ { 0 } } \frac { \partial L } { \partial \mathbf x ^ { \varepsilon } } \Biggr ) } & \\ & { = \frac { \partial ^ { \mathcal { W } _ { \varepsilon } ^ { 0 } } } { \partial \mathbf x } \mathrm { t a n } \left( \mathrm { r a n } _ { \mathcal { W } _ { \varepsilon } ^ { 0 } } ^ { \mathcal { W } _ { \varepsilon } ^ { 0 } } \frac { \partial L } { \partial \mathbf x ^ { \varepsilon } } \right) } & { \qquad \mathrm { ( i n d i g e r e n u i n ~ s t a n ~ u n i n ) } } \\ & { = \frac { \partial ^ { \mathcal { W } _ { \varepsilon } ^ { 0 } } } { \partial \mathbf x } \mathrm { t a n } ^ { \mathcal { W } _ { \varepsilon } ^ { 0 } } \mathrm { y s t a n } ^ { \mathcal { W } _ { \varepsilon } ^ { 0 } } \Biggr ) } & { \qquad \times \mathrm { r a n } _ { \mathcal { W } _ { \varepsilon } ^ { 0 } } ^ { \mathcal { W } _ { \varepsilon } ^ { 0 } } \mathrm { s t a n } \mathrm { m a n } \mathrm { m a n } \mathrm { W a } \mathrm { W a } \mathrm { W a } \mathrm { W a } \mathrm { W a } \mathrm { W a } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } } \\ &  = \sum _ { m _ { 0 } } ^ { \mathcal { W } _ { \varepsilon } ^ { 0 } } \mathrm { r a n } _ { \mathcal { W } _ { \varepsilon } ^ { 0 } } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W } \mathrm { W }  \end{array}
+$$
+
+Assume $\mathrm { V a r } ( w _ { m j } ^ { [ \ell ] } ) { = } \mathrm { V a r } ( w _ { n j } ^ { [ \ell ] } ) , \forall n , m$ , i.e. the variance of all weights for a given neuron are the same, and $\scriptstyle \mathrm { V a r } ( { \frac { \partial L } { \partial z _ { n } ^ { [ \ell ] } } } ) = \mathrm { V a r } ( { \frac { \partial L } { \partial z _ { m } ^ { [ \ell ] } } } )$ $l$ Then we can simplify Eq. (20),
+
+$$
+\begin{array} { r } { \mathrm { V a r } ( \displaystyle \frac { \partial L } { \partial z _ { j } ^ { \left[ \ell \right] } } ) = \displaystyle \sum _ { i = 1 } ^ { n ^ { \left[ \ell \right] } } m _ { i j } ^ { \left[ \ell \right] } \mathrm { V a r } ( w _ { i j } ^ { \left[ \ell \right] } ) \mathrm { V a r } ( \displaystyle \frac { \partial L } { \partial z _ { i } ^ { \left[ \ell \right] } } ) } \\ { = \mathrm { V a r } ( w _ { i j } ^ { \left[ \ell \right] } ) \mathrm { V a r } ( \displaystyle \frac { \partial L } { \partial z _ { i } ^ { \left[ \ell \right] } } ) \displaystyle \sum _ { i = 1 } ^ { n ^ { \left[ \ell \right] } } m _ { i j } ^ { \left[ \ell \right] } . } \end{array}
+$$
+
+$i$ $o u t _ { j } ^ { [ \ell ] }$ $\begin{array} { r } { { \bf { \Lambda } } _ { \cdot o u t _ { j } ^ { [ \ell ] } } = \sum _ { i = 1 } ^ { n ^ { [ \ell ] } } m _ { i j } ^ { [ \ell ] } } \end{array}$
+
+$$
+\begin{array} { r l } & { \quad \mathrm { V a r } ( \displaystyle \frac { \partial L } { \partial z _ { j } ^ { [ \ell ] } } ) { = } f a n { \ - } o u _ { j } ^ { [ \ell ] } \mathrm { V a r } ( w _ { i j } ^ { [ \ell ] } ) \mathrm { V a r } ( \displaystyle \frac { \partial L } { \partial z _ { i } ^ { [ \ell ] } } ) } \\ & { \quad \mathrm { R e c a l l , V a r } ( \displaystyle \frac { \partial L } { \partial z _ { j } ^ { [ \ell ] } } ) { = } \mathrm { V a r } ( \displaystyle \frac { \partial L } { \partial z _ { i } ^ { [ \ell ] } } ) } \\ & { \quad \quad \quad \quad \Rightarrow \mathrm { V a r } ( w _ { i j } ^ { [ \ell ] } ) { = } \displaystyle \frac { 1 } { f a n { - } o u _ { j } ^ { [ \ell ] } } . } \end{array}
+$$
+
+Therefore, in order to have the output of each neuron $a _ { i } ^ { [ \ell ] }$ in layer $\ell$ to have unit variance, and mean 0, we need to sample the weights for each neuron from the normal distribution,
+
+$$
+[ w _ { i j } ^ { [ \ell ] } ] \sim { \cal N } \left( 0 , \frac { 1 } { f a n { - } i n _ { i } ^ { [ \ell ] } } \right) ,
+$$
+
+where $s _ { i } ^ { \ell }$ is the sparsity of weights of the neuron with output $a _ { i }$ . For the ReLU activation function, following the derivation in $\mathrm { H e }$ et al. (2015),
+
+$$
+[ w _ { i j } ^ { [ \ell ] } ] \sim { \cal N } \left( 0 , \frac { 2 } { f a n { - } i n _ { i } ^ { [ \ell ] } } \right) .
+$$
+
+Table 3: §4.3: Experiment Details/Hyperparameters. Initial Learning Rate (LR), LR Schedule (Sched.), Batchsize (Batch.), Momentum $( m )$ , Weight Decay (WD), $t _ { \mathrm { s t a r t } }$ , $t _ { \mathrm { e n d } }$ and $f$ are the pruning starting iteration, end iteration, and mask update frequency respectively.   
+
+<table><tr><td>Dataset</td><td>Model</td><td>total</td><td>Epochs</td><td>Batch.</td><td>LR</td><td>Sched.</td><td>m</td><td>WD</td><td>Sparsity</td><td colspan="3">Pruning</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>tstart</td><td>tend</td><td>f</td></tr><tr><td>MNIST</td><td>LeNet5</td><td>11719</td><td>30</td><td>128</td><td>0.05</td><td>Cosine</td><td>0.9</td><td>0</td><td>95%</td><td>3000</td><td>7000</td><td>100</td></tr><tr><td>ImageNet</td><td>ResNet50</td><td>32000</td><td>~102</td><td>4096</td><td>1.6</td><td>Step</td><td>0.9</td><td></td><td>1×10-4 80%</td><td>5000</td><td>8000</td><td>2000</td></tr></table>
+
+\* Step schedule has a linear warm-up in first 5 epochs and decreases the learning rate by a factor of 10 at epochs 30,70 and 90.
+
+Table 4: $\ S 4 . 1$ : Experiment Details/Hyperparameters. Initial Learning Rate (LR), LR Schedule (Sched.), Batchsize (Batch.), Momentum $( m )$ , Weight Decay (WD), Initial Drop Fraction (Drop.), $t _ { \mathrm { e n d } }$ and $f$ are the pruning mask update frequency and end iteration respectively. $L e N e t 5 +$ row corresponds the LeNet5 experiments with our sparse initialization, whereas LeNet5 is the regular masked initialization.   
+
+<table><tr><td>Dataset</td><td>Model</td><td>total</td><td>Epochs</td><td>Batch.</td><td>LR</td><td>Sched.</td><td>m</td><td>WD</td><td>Sparsity</td><td>DST Drop.</td><td>f</td><td>tend</td></tr><tr><td>MNIST</td><td>LeNet5</td><td>11719</td><td>30</td><td>128</td><td>0.05</td><td>Cosine 0.9</td><td></td><td></td><td>5×10-4 95%</td><td>0.3</td><td>500</td><td>11719</td></tr><tr><td></td><td>ResNet50</td><td>）32000</td><td></td><td>4096</td><td>1.6</td><td></td><td></td><td></td><td></td><td>0.3</td><td>100</td><td></td></tr><tr><td>ImageNet</td><td>VGG16</td><td>128000</td><td>~102</td><td>1024</td><td>0.04</td><td>Step</td><td>0.9</td><td></td><td>1×10-4 80%</td><td>0.1</td><td>500</td><td>25000</td></tr></table>
+
+\* Step schedule has a linear warm-up in first 5 epochs and decreases the learning rate by a factor of 10 at epochs 30,70 and 90.
+
+# B Experimental Details
+
+# B.1 Details of Experiments in Section 4.3
+
+The training hyper-parameters used in $\ S 4 . 3$ are shared in Table 3. All experiments in this section start with a pruning experiment, after which the sparsity masks found by pruning are used to perform LT experiments. We use iterative magnitude pruning (Zhu et al., 2018) in our experiments, which is a well studied and more efficient pruning method as compared to the one used by Frankle et al. (2019a). Our pruning algorithm performs iterative pruning without rewinding the weights between intermediate steps and requires significantly less iterations. We expect our results would be even more pronounced with additional rewinding steps.
+
+We use SGD with momentum in all of our experiments. Scratch and Lottery experiments use the same hyper-parameters. Additional specific details of our experiments are shared below.
+
+LeNet5 We prune all layers of LeNet5, so that they reach $9 5 \%$ final sparsity (i.e. $9 5 \%$ of the parameters are zeros). We choose this sparsity, since at this sparsity, we start observing stark differences between Lottery and Scratch in terms of performance. We set the weight decay to zero, similar to the MNIST experiments done in the original LT paper (Frankle et al., 2019a) and do a grid search over learning-rates={0.1,0.2,0.05,0.02,0.01}. Loss values for the linear interpolation experiments are calculated on the entire training set.
+
+ResNet50 We prune all layers of ResNet50, except the first layer, so that they reach $80 \%$ final sparsity. In this setting rewinding to the original initialization doesn’t work, hence we use values from $6 ^ { \mathrm { { t h } } }$ epoch. Loss values for the linear interpolation experiments are calculated using 500,000 images from the ImageNet-2012 training set.
+
+# B.2 Details of Experiments in Section 4.1 and 4.2
+
+Training hyper-parameters used for these experiments are shared in Table 4.
+
+MNIST In this setting, the hyper-parameters are almost same as in $\ S 4 . 3$ , except we enable weight decay as it brings better generalization. We do a grid search over weightdecays={0.001,0.0001,0.00005,0.00001,0.0005} and learning-rates $= \{ 0 . \bar { 1 } , 0 . 2 , 0 . 0 5 , 0 . 0 2 , 0 . 0 1 \}$ and pick the values with top test accuracy. We use the masks found by pruning experiments in all of our MNIST experiments in this section to isolate the effect of the initialization. We simplify the update schedule of Dynamic Sparse Training (DST) methods such that they decay with learning rate. This approach fits well, since the original decay function used in these experiments is the cosine decay which is the same as our learning rate schedule. We scale learning rate such that it matches the initial drop fraction provided. Mask update frequency and initial drop fraction are chosen from a grid search o $\left\{ 5 0 , 1 0 0 , 5 0 0 \right\}$ and $\{ 0 . 0 1 , 0 . 1 , 0 . 3 \}$ respectively. To allow fair comparison, we use Glorot scaling in all of our initializations (i.e. scal $^ { \mathrm { { = 1 } } }$ and we average fan-in fan-out values) as it is the default initialization for Tensorflow layers and our results shows that it out-performs He initialization by a small margin with the hyper-parameters used. Using He initialization brings similar results.
+
+![](images/ecaea9e2629e74ce8366d78ec39c21d0330f0b8d2ae7c82d4f29bfb9b51bf1fd.jpg)  
+Figure 7: Gradient Flow of Sparse LeNet-5s during Training. Gradient flow during training averaged over multiple runs, $\cdot _ { + } \cdot$ indicates training runs with our proposed sparse initialization. $\cdot _ { \mathrm { + L i u } } \cdot$ indicates initialization proposed by Liu et al., 2019. ‘He‘ suffix refers to He initilization where ‘scale $= 2 ^ { \circ }$ and ‘fanin‘ options are used for the variance scaling initialization.
+
+Hessian calculation The Hessian is calculated on full training set using Hessian-vector products. We mask our network after each gradient call and calculate only non-zero rows. After calculating the full Hessian, we use numpy.eigh (van der Walt et al., 2011) to calculate eigenvalues of the Hessian.
+
+ImageNet-2012 In this setting, hyper-parameters are almost the same as in $\ S 4 . 3$ except for VGG16 architecture, where we use a smaller batch size and learning rate. For all DST methods, we use a cosine drop schedule Dettmers et al., 2019 and hyper-parameters proposed by Evci et al. (2019). For VGG, we reduce the mask update frequency and the initial drop fraction, as we observe better performance after doing a grid search over $\{ 5 0 , 1 0 0 , 5 0 0 \}$ and $\{ 0 . 1 , 0 . 3 , 0 . 5 \}$ respectively. We also use a non-uniform (ERK) sparsity distribution among layers as described in Evci et al. (2020), since we observed that it brings better performance.
+
+# C Additional Gradient Flow Plots
+
+Here we share additional gradient flow figures for method/initialization combinations presented in 3 and 2.
+
+In Fig. 7b, we show the gradient flow for DST methods and scratch training. Using RigL helps improves gradient flow with both initialization; helping learning to start earlier than regular Scratch training. Sparse Evolutionary Training (SET) seem to have limited effect on the gradient flow.
+
+In Fig. 7b, we share gradient flow when the scaled initialization of Liu et al., 2019 is used. Similar to the proposed initialization, we observe improved gradient flow for all cases. Different than our initialization however, RigL doesn’t improve gradient flow in this setting; highlighting an interesting future research direction on the relationship between initialization and the DST methods.
+
+In Fig. 7c, we share gradient flow when He initialization is used instead of Glorot initialization. We observe that Scratch training starts learning faster in this case. Gradient flow seems to be similar for other sparse initialization methods.
+
+In Fig. 8, we share gradient flow improvements after DST updates on connectivity for different initialization methods. ResNet-50 curves match the results in Fig. 3. LeNet5 curves however seem to be adversely affected by poor initialization at the beginning of the training. We start observing improvements with RigL when the learning starts (around the $4 0 0 ^ { \mathrm { { t h } } }$ iteration).
+
+![](images/1e418cdeac05d8e7522c3ea5a7ce1ea47adac00a56b152f700b88cafd28e15f1.jpg)  
+Figure 8: Effect of Mask Updates in Dynamic Sparse Training. Effect of mask updates on the gradient norm. We measure the gradient norm before and after the mask updates and plot the $\Delta$ . $^ \bullet + ^ { \bullet }$ indicates proposed initialization and used in MNIST experiments.
+
+Table 5: Results of Trained Fully-Connected MNIST Model from Different Initializations.   
+
+<table><tr><td rowspan="5">Baseline Lottery Small Dense</td><td colspan="3">MNIST</td></tr><tr><td colspan="3">Fully-Connected NN (98% sparse)</td></tr><tr><td colspan="3">98.55±0.04 97.73±0.11 91.69±1.85</td></tr><tr><td colspan="3">Original Liu et al.</td></tr><tr><td>94.40±4.00</td><td></td><td>Ours</td></tr><tr><td>Scratch</td><td></td><td>96.66±0.18</td><td>96.70±0.12</td></tr><tr><td>SET</td><td>96.49±0.36</td><td>96.56±0.22</td><td>96.48±0.10</td></tr><tr><td>RigL</td><td>96.82±0.25</td><td>96.76±0.19</td><td>96.94±0.12</td></tr></table>
+
+# D Fully Connected Neural Network Experiments on MNIST
+
+In this section we repeat our experiments from $\ S 4 . 1$ using different sparse initialization methods, and analyzing gradient flow, for a standard 2-layer fully-connected NN with 2 hidden layers of size 300 and 100 units. We use the same grid used in LeNet5 experiments for hyper-parameter selection. Best results were obtained with a learning rate of 0.2, a weight decay coefficient of 0.0001 and an mask update frequency of 500 (used in DST methods). The rest of the hyperparameters remained unchanged from the LeNet5 experiments.
+
+The results of training with various initialization methods is shown in Table 5. Although the results are not as drastic as with LeNet5, we see that here too sparsity aware initialization (the proposed initialization, and that of Liu (Liu et al., 2019)) shows a significant improvement in the test accuracy of Scratch, and RigL or our proposed initialization, although not quite reaching lottery or RigL accuracy. Finally, we see no significant effect on SET training, with none of the initialization variants having a significant increase over any of the others, although the Liu (Liu et al., 2019) initialization does marginally better.
+
+The gradient flow of this model is shown in Fig. 9. While we see moderate improvements to Scratch gradient flow early on in training with our proposed initialization (a), RigL shows significantly higher gradient flow throughout training, in particular after mask updates (b), mirroring the results of LeNet5. The interpolation graphs in (c) only differ slightly from that of LeNet5, again showing that our results for LeNet5 broadly hold for the fully-connected model.
+
+# E Hessian Spectrum of LeNet5
+
+Given a loss function $L$ and parameters $\theta$ , we can write the first order Taylor approximation of the change in loss $\Delta { \cal L } = { \cal L } ( \theta ^ { t + 1 } ) - { \cal L } ( \theta ^ { \hat { t } } )$ after a single training step with the learning rate $\epsilon > 0$ as :
+
+$$
+\Delta L { \approx } - \epsilon { \boldsymbol { \nabla } } L ( { \boldsymbol { \theta } } ) ^ { T } { \boldsymbol { \nabla } } L ( { \boldsymbol { \theta } } ) .
+$$
+
+Note that as long as the error is small, gradient descent is guaranteed to decrease the loss by an amount proportional to $\check { \nabla } L ( \theta ) ^ { T } \nabla L ( \theta )$ , which we refer as the gradient flow. In practice large learning rates are used, and the first order approximation might not be accurate. Instead we can look at the second order
+
+![](images/95bd693b1493826f6d9fc99d8e0506cdf1579c7440f44fda2ddb6e37e059dbb1.jpg)  
+Figure 9: Sparse 300-100 MLP experiments. Gradient flow during training averaged over multiple runs, $^ { \bullet } + ^ { \bullet }$ indicates training runs with our proposed sparse initialization.
+
+approximation of $\Delta L$
+
+$$
+\Delta L \approx - \alpha \boldsymbol { \nabla } L ( \theta ) ^ { T } \boldsymbol { \nabla } L ( \theta ) + \frac { \alpha ^ { 2 } } { 2 } \boldsymbol { \nabla } L ( \theta ) ^ { T } \boldsymbol { H } ( \theta ) \boldsymbol { \nabla } L ( \theta ) ,
+$$
+
+where $H ( \theta )$ is the Hessian of the loss function. The eigenvalue spectrum of Hessian can help us understand the local landscape (Sagun et al., 2017), and help us identify optimization difficulties (Ghorbani et al., 2019). For example, if and when the gradient is aligned with large magnitude eigenvalues, the second term of Eq. (28) can have a significant effect on the optimization of $L$ . If the gradient is aligned with large positive eigenvalues, it can prevent gradient descent from decreasing the loss and harm the optimization. Similarly, if it is aligned with negative eigenvalues it can help to accelerate optimization.
+
+We show the Hessian spectrum before and after the topology updates in Fig. 10. After RigL updates we observe new negative eigenvalues with significantly larger magnitudes. We also see larger positive eigenvalues, which disappear after few iterations\*\*. In comparison, the effect of SET updates on the Hessian spectrum seems limited.
+
+We also evaluate the Hessian spectrum of LeNet5 during the training. In Fig. 11b, we observe similar shapes for each method on the positive side of the spectrum, however, on the negative side dense models seem to have more mass. We plot the magnitude of the largest negative eigenvalue to characterize this behaviour in Fig. 11a. We observe a significant difference between sparse and dense models and observe that sparse networks trained with RigL have larger negative eigenvalues.
+
+# F Comparing Function Similarity
+
+Table 6 gives a full list of comparison metrics of the predictions on the test set for LeNet5 on MNIST and ResNet50 on ImageNet-2012, in particular here we also compare the output probability distributions using relevant metrics.
+
+<table><tr><td rowspan="11">JPsur/mmnr Sps ppo- PITn ud / r TTr rrllrpriir/i lirtsr Tarr tsrtetritetrr-itrr rittts Piiernr</td><td>000.53333.0 010.107007.0 100.1717</td><td>000.0--/7.0.0 100.007:10.1</td><td rowspan="11">3:0033.83</td><td rowspan="11">pJun ud /m gsr P1n.dd / a P iiiTitr prrnrr/ aertegg Pirialing</td><td rowspan="3">3331.355 171211222 555357174</td><td rowspan="3">3733.2558 33333.7311 222441 111211155</td></tr><tr><td>1770:009.5545 1000.001010&#x27;0</td><td>0013.1077153 1sf ppoa-s</td></tr><tr><td>0000.0010010 000530 55500</td></tr><tr><td>8000.353335. 8000.007555.5 0000.503575.1</td><td>0000&#x27;0F6111.7 0007.3031151 0000:0-5101&#x27;0</td><td rowspan="2">572537731 60000-1/77.0 0005.333330.5</td></tr><tr><td>0000.-03005.4 900.001-71.7 0715.5535555</td><td>0010:001111.0 **6100026100 00000-/2.0.0</td></tr><tr><td>7000&#x27;0F6800&#x27;0 9000.-01500.0 0010.157515.5 00：3.355703.3</td><td>0070007100.0 0000177105.5 0000.70720.0 002.105515.5</td></tr><tr><td>89.86 00&#x27;86 85</td><td>Prrrsr ssersee Jrreeteg £9&#x27;86 ∠0&#x27;66</td></tr><tr><td>70:3257.86 1:00 40:27 130035.33 10&#x27;0干 09&#x27;86 685 uos piunnm</td><td>Jr-1I) &#x27;eSt 1Ss 171.5330100 50154 ThrErNe</td></tr><tr><td>u H/M pend er Jia niers Prrereeere Sitett I</td><td>111 7511 11315751 Ja/M pand Shteett</td></tr></table>
+
+  
+g          . runinsolutionthattheLTandscratchmodelsared
+
+![](images/0008a38227fe13a4d42cd90d21735ab859396e37cd3139fc1f6b0a8cd0fc3ced.jpg)  
+Figure 10: Hessian spectrum before and after mask updates: (left) SET (right) RigL. Similar to Ghorbani et al., 2019, we estimate the spectral density of Hessian using Gaussian kernels.
+
+![](images/4397ebd086c96142f7220d78e3c6dfecc192dd075db0cbeb142ab552c67a8521.jpg)  
+Figure 11: MNIST Hessian spectrum experiments.

parse/train/iorEu783qJ5/iorEu783qJ5.md

@@ -0,0 +1,210 @@
+# Particle Cloud Generation with Message Passing Generative Adversarial Networks
+
+Raghav Kansal, Javier Duarte, Hao Su University of California San Diego La Jolla, CA 92093, USA
+
+Breno Orzari, Thiago Tomei Universidade Estadual Paulista São Paulo/SP - CEP 01049-010, Brazil
+
+Maurizio Pierini, Mary Touranakou⇤ European Organization for Nuclear Research (CERN) CH-1211 Geneva 23, Switzerland
+
+Jean-Roch Vlimant California Institute of Technology Pasadena, CA 91125, USA
+
+Dimitrios Gunopulos National and Kapodistrian University of Athens Athens 15772, Greece
+
+# Abstract
+
+In high energy physics (HEP), jets are collections of correlated particles produced ubiquitously in particle collisions such as those at the CERN Large Hadron Collider (LHC). Machine learning (ML)-based generative models, such as generative adversarial networks (GANs), have the potential to significantly accelerate LHC jet simulations. However, despite jets having a natural representation as a set of particles in momentum-space, a.k.a. a particle cloud, there exist no generative models applied to such a dataset. In this work, we introduce a new particle cloud dataset (JetNet), and apply to it existing point cloud GANs. Results are evaluated using (1) 1-Wasserstein distances between high- and low-level feature distributions, (2) a newly developed Fréchet ParticleNet Distance, and (3) the coverage and (4) minimum matching distance metrics. Existing GANs are found to be inadequate for physics applications, hence we develop a new message passing GAN (MPGAN), which outperforms existing point cloud GANs on virtually every metric and shows promise for use in HEP. We propose JetNet as a novel point-cloud-style dataset for the ML community to experiment with, and set MPGAN as a benchmark to improve upon for future generative models. Additionally, to facilitate research and improve accessibility and reproducibility in this area, we release the open-source JETNET Python package with interfaces for particle cloud datasets, implementations for evaluation and loss metrics, and more tools for ML in HEP development.
+
+# 1 Introduction
+
+Over the past decade, machine learning (ML) has become the de facto way to analyze jets, collimated high-energy sprays of particles [1] produced at the CERN Large Hadron Collider (LHC). To apply ML to jets, the most natural representation is a particle cloud, a variable-sized set of points in momentum space, whose radiation pattern contains rich information about the underlying physics known as quantum chromodynamics (QCD). A fundamental question is whether ML algorithms can model this underlying physics and successfully reproduce the rich high- and low-level structure in jets.
+
+Answering this question affirmatively has important practical applications. At the LHC, large simulated data samples of collision events2 are generated using Monte Carlo (MC) methods in order to translate theoretical predictions into observable distributions, and ultimately perform physics analyses3. These samples, numbering in the billions of events, require computationally expensive modeling of the interaction of particles traversing the detector material. Recently developed generative frameworks in ML such as generative adversarial networks (GANs), if accurate enough, can be used to accelerate this simulation by potentially five orders of magnitude [2].
+
+In this work, we advocate for a benchmark jet dataset (JetNet) and propose several physics- and computer-vision-inspired metrics with which the ML community can improve and evaluate generative models in high energy physics (HEP). To facilitate and encourage research in this area, as well as to make such research more accessible and reproducible, we release interfaces for public particle cloud datasets such as JetNet, implementations for our proposed metrics, and various tools for ML in HEP development in the JETNET library [3]. We next apply existing point cloud GANs on JetNet and find the results to be inadequate for physics applications. Finally, we develop our own message passing GAN (MPGAN), which dramatically improves results on virtually every metric, and propose it as a benchmark on JetNet.
+
+# 2 Jets
+
+High-energy proton-proton collisions at the LHC produce elementary particles like quarks and gluons, which cannot be isolated due to the QCD property of color confinement [4]. These particles continuously radiate or “split” into a set of particles, known as a parton shower. Eventually they cool to an energy at which they undergo the process of hadronization, where the fundamental particles combine to form more stable hadrons, such as pions and protons. The final set of collimated hadrons produced after such a process is referred to as a jet.
+
+The task of simulating a single jet can be algorithmically defined as inputting an initial particle, which produces the jet, and outputting the final set of particles a.k.a. the jet constituents. Typically in HEP the parton shower and hadronization are steps that are simulated sequentially using MC event generators such as PYTHIA [5] or HERWIG [6]. Simulating either process exactly is not possible because of the complex underlying physics (QCD), and instead these event generators fit simplified physics-inspired stochastic models, such as the Lund string model for hadronization [7], to existing data using MC methods. The present work can be seen as an extension of this idea, using a simpler, ML-based model, also fitted to data, for generating the jet in one shot, where we are effectively trading the interpretability of the MC methods for the speed of GPU-accelerated ML generators.
+
+Representations. As common for collider physics, we use a Cartesian coordinate system with the $z$ axis oriented along the beam axis, the $x$ axis on the horizontal plane, and the $y$ axis oriented upward. The $x$ and $y$ axes define the transverse plane, while the $z$ axis identifies the longitudinal direction. The azimuthal angle $\phi$ is computed with respect to the $x$ axis. The polar angle $\theta$ is used to compute the pseudorapidity $\eta = - \log ( \tan ( \theta / 2 ) )$ . The transverse momentum $( p _ { \mathrm { T } } )$ is the projection of the particle momentum on the $( x , y )$ plane. As is customary, we transform the particle momenta from Cartesian coordinates $\left( p _ { x } , p _ { y } , p _ { z } \right)$ to longitudinal-boost-invariant pseudo-angular coordinates $( p _ { \mathrm { T } } , \eta , \phi )$ , as shown in Fig. 1.
+
+In the context of ML, jets can be represented in multiple ways. One popular representation is as images [8, 9], created by projecting each jet’s particle constituents onto a discretized angular $\eta { - } \phi$ plane, and taking the intensity of each “pixel” in this grid to be a monotonically increasing function of the corresponding particle $p _ { \mathrm { T } }$ . These tend to be extremely sparse, with typically fewer than $10 \%$ of pixels nonempty [10], and the discretization process can furthemore lower the resolution.
+
+Two more spatially efficient representations are as ordered lists or unordered sets [11, 12] of the jet constituents and their features. The difficulty with the former is that there is no particular preferred ordering of the particles—one would have to impose an arbitrary ordering such as by transverse momentum [13]. The more natural representation is the unordered set of particles in momentum space, which we refer to as a “particle cloud.” This is in analogy to point cloud representations of 3D objects in position-space prevalent in computer vision created, for example, by sampling from 3D ShapeNet models [14].
+
+![](images/84898c4b21c9c3390cd9178cd2c4e55c4ea0b64586f4d8d3d28bd30f3b2d3275.jpg)  
+Figure 1: The collider physics coordinate system defining $( p _ { \mathrm { T } } , \eta , \phi )$ (left). The three jet classes in our dataset (right). Gluon (g) and light quark (q) jets have simple topologies, with q jets generally containing fewer particles. Top quark (t) jets have a complex three-pronged structure. Shown also are the relative angular coordinates $\bar { \boldsymbol { \eta } } ^ { \mathrm { r e l } }$ h→and $\dot { \phi } ^ { \mathrm { r e l } }$ t→Wb→q, measured from the jet axis.
+
+Apart from how the samples are produced, significant differences between jets and ShapeNet-based point clouds are that, firstly, jets have physically meaningful low- and high-level features such as particle momentum, total mass of the jet, the number of sub-jets, and $n$ -particle energy correlations. These physical observables are how we characterize jets, and hence are important to reproduce correctly for physics analysis applications. Secondly, unlike the conditional distributions of points given a particular ShapeNet object, which are identical and independent, particle distributions within jets are highly correlated, as the particles each originate from a single source. The independence of their constituents also means that ShapeNet-sampled point clouds can be chosen to be of a fixed cardinality, whereas this is not possible for jets, which inherently contain varying numbers of particles due to the stochastic nature of particle production.
+
+JetNet. We publish JetNet [15] under the CC-BY 4.0 license, to facilitate and advance ML research in HEP, and to offer a new point-cloud-style dataset to experiment with. Derived from Ref. $[ 1 6 ] ^ { 4 }$ , it consists of simulated particle jets with transverse momenta $p _ { \mathrm { T } } ^ { \mathrm { j e t } } \approx 1 \mathrm { T e V }$ , originating from gluons, light quarks, and top quarks produced in $1 3 \mathrm { T e V }$ proton-proton collisions in a simplified detector. Technical details of the generation process are given in App. B. We limit the number of constituents to the 30 highest $p _ { \mathrm { T } }$ particles per jet, allowing for jets with potentially fewer than 30 by zeropadding. For each particle we provide the following four features: the relative angular coordinates $\bar { \eta } ^ { \mathrm { r e l } } = \bar { \eta } ^ { \mathrm { p a r t i c l e } } - \bar { \eta } ^ { \mathrm { j e t } }$ and $\phi ^ { \mathrm { r e l } } { \bar { = } } \phi ^ { \mathrm { p a r t i c l e } } - \phi ^ { \mathrm { j e t } }$ (mod $2 \pi$ ), relative transverse momentum $p _ { \mathrm { T } } ^ { \mathrm { r e l } } =$ $p _ { \mathrm { T } } ^ { \mathrm { p a r t i c l e } } / p _ { \mathrm { T } } ^ { \mathrm { j e t } }$ , and a binary mask feature classifying the particle as genuine or zero-padded.
+
+We choose three jet classes, depicted in Fig. 1, to individually target the unique and challenging properties of jets. Gluons provide a useful baseline test, as they typically radiate into a large number of particles before hadronization, largely avoiding the variable-sized cloud issue—at least with a 30 particle maximum, and have a relatively simple topology. Light quarks share the simple topology, but produce fewer final-state particles, resulting in a larger fraction of zero-padded particles in the dataset. They allow evaluation of a model’s ability to handle variable-sized clouds. Finally, top quarks decay into three lighter quarks through an intermediate particle, the W boson, which each may produce their own sub-jets, leading to a complex two- or three-pronged topology—depending on whether the jet clustering algorithm captures all three or just two of these sub-jets. This results in bimodal jet feature distributions (one peak corresponding to fully merged top quark jets and the other to semi-merged, as seen in Fig. 3). Thus, top quark jets test models’ ability to learn the rich global structure and clustering history of a particle cloud.
+
+# 3 Related Work
+
+Generative models in HEP. Past work in this area has exclusively used image-based representations for HEP data. One benefit of this is the ability to employ convolutional neural network (CNN) based generative models, which have been highly successful on computer vision tasks.
+
+Refs. [2, 17–20], for example, build upon CNN-based GANs, and Ref. [21] uses an auto-regressive model, to output jet- and detector-data-images.
+
+In addition to the issues with such representations outlined in Sec. 2, the high sparsity of the images can lead to training difficulties in GANs, and the irregular geometry of the data — a single LHC detector can typically have multiple sections with differing pixel sizes and shapes — poses a challenge for CNN GANs which output uniform matrices. While these can be mitigated to an extent with techniques such as batch normalization [22] and using larger/more regular pixels [18], our approach avoids both issues by generating particle-cloud-representations of the data, as these are inherently sparse data structures and are completely flexible to the underlying geometry.
+
+GANs for point clouds. There are several published generative models in this area, however the majority exploit inductive biases specific to their respective datasets, such as ShapeNet-based [23–26] and molecular [27–29] point clouds, which are not appropriate for jets. A more detailed discussion, including some experimental results, can be found in App. C.
+
+There do exist some more general-purpose GAN models, namely r-GAN [30], GraphCNN-GAN [31], and TreeGAN [32], and we test these on JetNet. r-GAN uses a fully-connected (FC) network, GraphCNN-GAN uses graph convolutions based on dynamic $k$ -nn graphs in intermediate feature spaces, and TreeGAN iteratively up-samples the graphs with information passing from ancestor to descendant nodes. In terms of discriminators, past work has used either a FC or a PointNet [33]-style network. Ref. [34] is the first work to study point cloud discriminator design in detail and finds amongst a number of PointNet and graph convolutional models that PointNet-Mix, which uses both max- and average-pooled features, is the most performant.
+
+We apply the three aforementioned generators and FC and PointNet-Mix discriminators as baselines to our dataset, but find jet structure is not adequately reproduced. GraphCNN’s local convolutions make learning global structure difficult, and while the TreeGAN and FC generator $^ +$ PointNet discriminator combinations are improvements, they are not able to learn multi-particle correlations, particularly for the complex top quark jets, nor deal with the variable-sized light quark jets to the extent necessary for physics applications.
+
+Message Passing Neural Networks. We attempt to overcome limitations of existing GANs by designing a novel generator and discriminator which can learn such correlations and handle variablesized particle clouds. Both networks build upon the generic message-passing neural network (MPNN) [35] framework with physics-conscious design choices, and collectively we refer to them as message-passing GAN (MPGAN). We find MPGAN outperforms existing models on virtually all evaluation metrics.
+
+# 3.1 Evaluating generative models.
+
+Evaluating generative models is a difficult task, however there has been extensive work in this area in both the physics and computer-vision communities.
+
+Physics-inspired metrics. An accurate jet simulation algorithm should reproduce both low-level and high-level features (such as those described in Sec. 2), hence a standard method of validating generative models, which we employ, is to compare the distributions of such features between the real and generated samples5 [2, 17–20, 36].
+
+For application in HEP, a generative model needs to produce jets with physical features indistinguishable from real. Therefore, we propose the validation criteria that differences between real and generated sample features may not exceed those between sets of randomly chosen real samples. To verify this, we use bootstrapping to compare between random samples of only real jets as a baseline.
+
+A practically useful set of features to validate against are the so-called “energy-flow polynomials” (EFPs) [37], which are a set of multi-particle correlation functions. Importantly, the set of all EFPs forms a linear basis for all useful jet-level features6. Therefore, we claim that if we observe all EFP distributions to be reproduced with high fidelity and to match the above criteria, we can conclude with strong confidence that our model is outputting accurate particle clouds.
+
+![](images/5fece647d1585c77642972c14b6d56ef8139794d5fd92b6e0f81e0f4bbc17a39.jpg)  
+Figure 2: Top: The MP generator uses message passing to generate a particle cloud. In blue is the initial latent vector and FC layer part of the MP-LFC variant. Bottom: The MP discriminator uses message passing to classify an input particle cloud as real or generated.
+
+Computer-vision-inspired metrics A popular metric for evaluating images which has shown to be sensitive to output quality and mode-collapse, though it has its limitations [38], is the Fréchet Inception Distance [39] (FID). FID is defined as the Fréchet distance between Gaussian distributions fitted to the activations of a fully-connected layer of the Inception-v3 image classifier in response to real and generated samples. We develop a particle-cloud-analogue of this metric, which we call Fréchet ParticleNet Distance (FPND), using the state-of-the-art (SOTA) ParticleNet graph convolutional jet classifier [10] in lieu of the Inception network. We note that the FPND and comparing distributions as above is conceptually equivalent, except here instead of physically meaningful and easily interpretable features, we are comparing those found to be statistically optimum for distinguishing jets.
+
+Two common metrics for evaluating point cloud generators are coverage (COV) and minimum matching distance (MMD) [30]. Both involve finding the closest point cloud in a sample $X$ to each cloud in another sample $Y$ , based on a metric such as the Chamfer distance or the earth mover’s distance. Coverage is defined as the fraction of samples in $X$ which were matched to one in $Y$ , measuring thus the diversity of the samples in $Y$ relative to $X$ , and MMD is the average distance between matched samples, measuring the quality of samples. We use both, and due to drawbacks of the Chamfer distance pointed out in Ref. [30], for our distance metric choose only the analogue of the earth mover’s distance for particle clouds a.k.a. the energy mover’s distance (EMD) [40]. We discuss the effectiveness and complementarity of all four metrics in evaluating clouds in Sec. 5.
+
+# 4 MPGAN Architecture
+
+We describe now the architecture of our MPGAN model (Fig. 2), noting particle cloud-motivated aspects compared to its r-GAN and GraphCNN-GAN predecessors.
+
+Message passing. Jets originate from a single source particle decaying and hadronizing, hence they end up with important high-level jet features and a rich global structure, known as the jet substructure [1], stemming from the input particle. Indeed any high-level feature useful for analyzing jets, such as jet mass or multi-particle correlations, is necessarily global [37]. Because of this, while past work in learning on point clouds [10, 41, 42], including GraphCNN-GAN, has used a locally connected graph structure and convolutions for message passing, we choose a fully connected graph, equally weighting messages from all particles in the clouds. Rather than subtracting particle features for messages between particles, useful in graph convolutions to capture local differences within a neighborhood, the respective features are concatenated to preserve the global structure (the difference between particle features is also only physically meaningful if they are in the 4-vector representation of the Lorentz group). During the update step in the message passing we find it empirically beneficial to incorporate a residual connection to previous particle features.
+
+The operation can be described as follows. For an $N$ -particle cloud $J ^ { t } = \{ p _ { 1 } ^ { t } , \cdot \cdot \cdot , p _ { N } ^ { t } \}$ after $t$ iterations of message passing, with $t = 0$ corresponding to the original input cloud, each particle $p _ { i } ^ { t }$ is represented by features $\mathbf { h } _ { i } ^ { t }$ . One iteration of message passing is then defined as
+
+$$
+\begin{array} { r l } { { } } & { { \mathbf { m } _ { i j } ^ { t + 1 } = f _ { e } ^ { t + 1 } ( \mathbf { h } _ { i } ^ { t } \oplus \mathbf { h } _ { j } ^ { t } ) , } } \\ { { } } & { { \mathbf { h } _ { i } ^ { t + 1 } = f _ { n } ^ { t + 1 } ( \mathbf { h } _ { i } ^ { t } \oplus \displaystyle \sum _ { j \in J } \mathbf { m } _ { i j } ^ { t + 1 } ) , } } \end{array}
+$$
+
+where particl $\mathbf { m } _ { i j } ^ { t + 1 }$ id mesand vector sent from particle are arbitrary functions w $j$ to particle ich, in our $i$ , $\mathbf { h } _ { i } ^ { t + 1 }$ are the updated features of implemented as multilayer $i$ $f _ { e } ^ { t + 1 }$ $f _ { n } ^ { t + 1 }$   
+perceptrons (MLPs) with 3 FC layers.
+
+Generator. We test two initializations of a particle cloud for the MPGAN generator: (1) directly initializing the cloud with $N$ particles with $L$ randomly sampled features, which we refer to as the MP generator, and (2) inputting a single $Z$ -dimensional latent noise vector and transforming it via an FC layer into an $N \times L$ -dimensional matrix, which we refer to as the MP-Latent-FC (MP-LFC) generator. The MP-LFC uses a latent space which can intuitively be understood as representing the initial source particle’s features along with parameters to capture the stochasticity of the jet production process. Due to the complex nature of this process, however, we posit that this global, flattened latent space cannot capture the full phase space of individual particle features. Hence, we introduce the MP generator, which samples noise directly per particle, and find that it outperforms MP-LFC (Table 2).
+
+Discriminator. We find the MP generator, in conjunction with a PointNet discriminator, to be a significant improvement on every metric compared to FC and GraphCNN generators. However, the jet-level features are not yet reproduced to a high enough accuracy (Sec. 5). While PointNet is able to capture global structural information, it can miss the complex interparticle correlations in real particle clouds. We find we can overcome this limitation by incorporating message passing in the discriminator as well as in the generator. Concretely, our MP discriminator receives the real or generated cloud and applies MP layers to produce intermediate features for each particle, which are then aggregated via a feature-wise average-pooling operation and passed through an FC layer to output the final scalar feature. We choose 2 MP layers for both networks.
+
+Variable-sized clouds. In order to handle clouds with varying numbers of particles, as typical of jets, we introduce an additional binary “masking” particle feature classifying the particle as genuine or zero-padded. Particles in the zero-padded class are ignored entirely in the message passing and pooling operations. The MP generator adds mask features to the initial particle cloud, using an additional input of the size of the jet $N$ , sampled from the real distribution, before the message passing layers based on sorting in particle feature space. Ablation studies with alternative (as well as no) masking strategies are discussed in App. E.
+
+# 5 Experiments
+
+Evaluation. We use four techniques discussed in Sec. 3.1 for evaluating and comparing models. Distributions of physical particle and jet features are compared visually and quantitatively using the Wasserstein-1 $( W _ { 1 } )$ distance between them. For ease of evaluation, we report (1) the average scores of the three particle features $( W _ { 1 } ^ { \mathrm { P } } ) \eta ^ { \mathrm { r e l } }$ , $\phi ^ { \mathrm { r e l } }$ , and $p _ { \mathrm { T } } ^ { \mathrm { r e l } }$ , (2) the jet mass $( W _ { 1 } ^ { \mathrm { M } } )$ , and (3) the average of a subset of the $\mathrm { E F P s } ^ { 7 } ( W _ { 1 } ^ { \mathrm { E F P } } )$ , which together provide a holistic picture of the low- and high-level aspects of a jet. The $W _ { 1 }$ distances are calculated for each feature between random samples of 10,000 real and generated jets, and averaged over 5 batches. Baseline $W _ { 1 }$ distances are calculated between two sets of randomly sampled real jets with 10,000 samples each, and are listed for each feature in Table 1. The real samples are split 70/30 for training/evaluation. We train ParticleNet for classification on our dataset to develop the FPND metric. FPND is calculated between 50,000 random real and generated samples, based on the activations of the first FC layer in our trained model8. Coverage and MMD are calculated between 100 real and 100 generated samples, and averaged over 10 such batches. Implementations for all metrics are provided in the JETNET package [3].
+
+Table 1: $W _ { 1 }$ distances between real jet mass $( W _ { 1 } ^ { \mathrm { M } } )$ , averaged particle features $( W _ { 1 } ^ { \mathrm { P } } )$ , and averaged jet EFPs $( W _ { 1 } ^ { \mathrm { E F P } } )$ distributions calculated as a baseline, for three classes of jets.   
+
+<table><tr><td>Jet class</td><td>WM (x10-3)</td><td>WP (×10-3)</td><td>WEFP (×10-5)</td></tr><tr><td>Gluon</td><td>0.7 ± 0.2</td><td>0.44± 0.09</td><td>0.62 ± 0.07</td></tr><tr><td>Light quark</td><td>0.5 ± 0.1</td><td>0.5 ± 0.1</td><td>0.46 ± 0.04</td></tr><tr><td>Top quark</td><td>0.51 ± 0.07</td><td>0.55 ± 0.07</td><td>1.1 ± 0.1</td></tr></table>
+
+![](images/51b648485cf27e970a00bad6019c1f8ab5c89a8359435edd729031e5c52bafd2.jpg)  
+Figure 3: Comparison of real and generated distributions for a subset of jet and particle features. We use the best performing model for each of the FC, GraphCNN, TreeGAN, and MP generators, as per Table 2. Top: gluon jet features, Middle: light quark jets, Bottom: top quark jets.
+
+Results. On each of JetNet’s three classes, we test r-GAN’s FC, GraphCNN, and TreeGAN generators with rGAN’s FC and the PointNet-Mix discriminators, and compare them to MPGAN’s MP generator and discriminator models, including both MP and MP-LFC generator variations. Training and implementation details for each can be found in App. D, and all code in Ref. [43].
+
+We choose model parameters which, during training, yield the lowest $W _ { 1 } ^ { \mathrm { M } }$ score. This is because (1) $W _ { 1 }$ scores between physical features are more relevant for physics applications than the other three metrics, and (2) qualitatively we find it be a better discriminator of model quality than particle features or EFP scores. Table 2 lists the scores for each model and class, and Fig. 3 shows plots of selected feature distributions of real and generated jets, for the best performing FC, GraphCNN, TreeGAN, and MP generators. We also provide in App. F discretized images in the angular-coordinates-plane a.k.a “jet images”, however, we note that it is in general not easy to visually evaluate the quality of individual particle clouds, hence we focus on metrics and visualizations aggregated over batches of clouds. Overall we find that MPGAN is a significant improvement over the best FC, GraphCNN, and TreeGAN models, particularly for top and light quark jets. This is evident both visually and quantitatively in every metric, especially jet $W _ { 1 } s$ and FPND, with the exception of $W _ { 1 } ^ { \mathrm { P } }$ where only the FC generator and PointNet discriminator $\mathrm { F C } +$ PointNet) combination is more performant.
+
+Table 2: Six evaluation scores on different generator and discriminator combinations. Lower is better for all metrics except COV.   
+
+<table><tr><td rowspan="2">Jet class</td><td rowspan="2">Generator</td><td rowspan="2">Discriminator</td><td rowspan="2">WM (x10-3)</td><td rowspan="2">WP (×10-3)</td><td rowspan="2">WEFP (×10-5)</td><td rowspan="2">FPND</td><td rowspan="2">MMD</td></tr><tr><td>COV ↑</td></tr><tr><td rowspan="12">Gluon</td><td>FC</td><td>FC</td><td>18.3± 0.2</td><td>9.6± 0.4</td><td>8.5±0.5</td><td>176</td><td>0.24</td><td>0.045</td></tr><tr><td>GraphCNN</td><td>FC</td><td>2.6± 0.2</td><td>9.6 ±0.3</td><td>12±8</td><td>61</td><td>0.39</td><td>0.046</td></tr><tr><td>TreeGAN</td><td>FC</td><td>41.9 ± 0.3</td><td>69.3 ± 0.3</td><td>14.2 ± 0.8</td><td>355</td><td>0.19</td><td>0.130</td></tr><tr><td>FC</td><td>PointNet</td><td>1.3 ± 0.4</td><td>1.3± 0.2</td><td>1.5 ± 0.9</td><td>5.0</td><td>0.49</td><td>0.039</td></tr><tr><td>GraphCNN</td><td>PointNet</td><td>1.9 ±0.2</td><td>16±6</td><td>200 ±1000</td><td>7k</td><td>0.46</td><td>0.040</td></tr><tr><td>TreeGAN</td><td>PointNet</td><td>1.7 ± 0.1</td><td>4.0± 0.4</td><td>4±1</td><td>84</td><td>0.37</td><td>0.042</td></tr><tr><td>MP</td><td>MP</td><td>0.7± 0.2</td><td>0.9 ± 0.3</td><td>0.7±0.2</td><td>0.12</td><td>0.56</td><td>0.037</td></tr><tr><td>MP-LFC</td><td>MP</td><td>0.69 ± 0.07</td><td>1.8± 0.2</td><td>0.9 ±0.6</td><td>0.20</td><td>0.54</td><td>0.037</td></tr><tr><td>FC</td><td>MP</td><td>4.3 ± 0.3</td><td>21.1 ±0.2</td><td>9±1</td><td>368</td><td>0.11</td><td>0.085</td></tr><tr><td>GraphCNN</td><td>MP</td><td>2.5 ± 0.1</td><td>9.8± 0.2</td><td>13±8</td><td>61</td><td>0.38</td><td>0.048</td></tr><tr><td>TreeGAN</td><td>MP</td><td>2.4± 0.2</td><td>12±7</td><td>18±9</td><td>69</td><td>0.34</td><td>0.048</td></tr><tr><td>MP</td><td>FC</td><td>1.2 ± 0.2</td><td>3.7 ± 0.5</td><td>1.6 ± 0.8</td><td>39</td><td>0.44</td><td>0.040</td></tr><tr><td>MP</td><td>PointNet</td><td>1.3± 0.4</td><td>1.2 ± 0.4</td><td>4±2</td><td>18</td><td>0.53</td><td>0.036</td></tr><tr><td rowspan="14">Light quark</td><td>FC</td><td>FC</td><td>6.0±0.2</td><td>16.3 ± 0.9</td><td>3.9 ±0.6</td><td>395</td><td>0.18</td><td>0.053</td></tr><tr><td>GraphCNN</td><td>FC</td><td>3.5± 0.2</td><td>15.1 ± 0.4</td><td>10±50</td><td>100</td><td>0.25</td><td>0.038</td></tr><tr><td>TreeGAN</td><td>FC</td><td>31.5 ± 0.3</td><td>22.3±0.4</td><td>9.3 ± 0.4</td><td>176</td><td>0.06</td><td>0.055</td></tr><tr><td>FC</td><td>PointNet</td><td>3.1 ± 0.2</td><td>4.5± 0.4</td><td>2.3 ± 0.6</td><td>17</td><td>0.37</td><td>0.028</td></tr><tr><td>GraphCNN</td><td>PointNet</td><td>4±1</td><td>5.2±0.5</td><td>50k±100k</td><td>316</td><td>0.37</td><td>0.031</td></tr><tr><td>TreeGAN</td><td>PointNet</td><td>10.1 ± 0.1</td><td>5.7±0.5</td><td>4.1 ± 0.3</td><td>11</td><td>0.47</td><td>0.031</td></tr><tr><td>MP</td><td>MP</td><td>0.6±0.2</td><td>4.9 ± 0.5</td><td>0.7± 0.4</td><td>0.35</td><td>0.50</td><td>0.026</td></tr><tr><td>MP-LFC</td><td>MP</td><td>0.7±0.2</td><td>2.6 ± 0.4</td><td>0.9 ± 0.9</td><td>0.08</td><td>0.52</td><td>0.024</td></tr><tr><td>FC</td><td>MP</td><td>6.3± 0.2</td><td>16.5 ± 0.2</td><td>4.0±0.8</td><td>212</td><td>0.11</td><td>0.070</td></tr><tr><td>GraphCNN</td><td>MP</td><td>3.5± 0.4</td><td>15.0 ± 0.3</td><td>10±10</td><td>99</td><td>0.26</td><td>0.038</td></tr><tr><td>TreeGAN</td><td>MP</td><td>4.8±0.2</td><td>33±6</td><td>10±2</td><td>148</td><td>0.22</td><td>0.041</td></tr><tr><td>MP</td><td>FC</td><td>1.3± 0.1</td><td>4.5± 0.4</td><td>2.2 ±0.6</td><td>41</td><td>0.37</td><td>0.030</td></tr><tr><td>MP</td><td>PointNet</td><td>6.5± 0.3</td><td>23.2±0.6</td><td>6±1</td><td>850</td><td>0.18</td><td>0.034</td></tr><tr><td></td><td>FC</td><td></td><td></td><td></td><td></td><td>0.28</td><td>0.103</td></tr><tr><td rowspan="14">Top quark</td><td>FC GraphCNN</td><td></td><td>4.8±0.3</td><td>14.5 ± 0.6</td><td>23±3</td><td>160</td><td></td><td>0.081</td></tr><tr><td>TreeGAN</td><td>FC</td><td>7.0±0.3</td><td>8.0±0.5</td><td>1k ±6k</td><td>15</td><td>0.48</td><td></td></tr><tr><td></td><td>FC</td><td>17.0 ± 0.2</td><td>19.6± 0.6</td><td>33±2</td><td>77</td><td>0.39</td><td>0.083</td></tr><tr><td>FC GraphCNN</td><td>PointNet PointNet</td><td>2.7± 0.1</td><td>1.6 ± 0.4</td><td>7.7 ±0.5</td><td>3.9</td><td>0.56</td><td>0.075 0.085</td></tr><tr><td>TreeGAN</td><td>PointNet</td><td>11.3 ± 0.9 5.19 ± 0.08</td><td>30±10 9.1 ± 0.3</td><td>37±2</td><td>30k 17</td><td>0.39 0.53</td><td>0.079</td></tr><tr><td>MP</td><td></td><td></td><td></td><td>16±2</td><td></td><td></td><td></td></tr><tr><td>MP-LFC</td><td>MP</td><td>0.6±0.2</td><td>2.3±0.3</td><td>2±1</td><td>0.37</td><td>0.57</td><td>0.071</td></tr><tr><td></td><td>MP</td><td>0.9±0.3</td><td>2.2±0.7</td><td>2±1</td><td>0.93</td><td>0.56</td><td>0.073</td></tr><tr><td>FC</td><td>MP</td><td>6.9 ± 0.1</td><td>39.1± 0.3</td><td>15±1</td><td>81</td><td>0.26</td><td>0.120</td></tr><tr><td>GraphCNN</td><td>MP</td><td>6.7±0.1</td><td>8.2±0.5</td><td>40±10</td><td>15</td><td>0.49</td><td>0.081</td></tr><tr><td>TreeGAN</td><td>MP</td><td>13.4 ± 0.4</td><td>45±7</td><td>50±30</td><td>66</td><td>0.29</td><td>0.101</td></tr><tr><td>MP</td><td>FC</td><td>12.9 ± 0.3</td><td>26.3± 0.4</td><td>46±3</td><td>58</td><td>0.27</td><td>0.103</td></tr><tr><td>MP</td><td>PointNet</td><td>0.76±0.08</td><td>1.6 ± 0.4</td><td>4±1</td><td>3.7</td><td>0.59</td><td>0.072</td></tr></table>
+
+We additionally perform a latency measurement and find, using an NVIDIA A100 GPU, that MPGAN generation requires $3 5 . 7 \mu \mathrm { s }$ per jet. In comparison, the traditional generation process for JetNet is measured on an 8-CPU machine as requiring 46ms per jet, meaning MPGAN provides a three-ordersof-magnitude speed-up. Furthermore, as noted in App. B, the generation of JetNet is significantly simpler than full simulation and reconstruction used at the LHC, which has been measured to require 12.3s [44] and 4s [45] respectively per top quark jet. Hence in practical applications we anticipate MPGAN’s improvement to potentially rise to five-orders-of-magnitude.
+
+Real baseline comparison. We find that MPGAN’s jet-level $W _ { 1 }$ scores all fall within error of the baselines in Table 1, while those of alternative generators are several standard deviations away. This is particularly an issue with complex top quark particle clouds, where we can see in Fig. 3 none of the existing generators are able to learn the bimodal jet feature distributions, and smaller light quark clouds, where we see distortion of jet features due to difficulty reproducing the zero-padded particle features. No model is able to achieve particle-level scores close to the baseline, and only those of the $\mathrm { F C } +$ PointNet combination and MPGAN are of the same order of magnitude. We conclude that MPGAN reproduces the physical observable distributions to the highest degree of accuracy, but note, however, that it requires further improvement in particle feature reconstruction before it is ready for practical application in HEP.
+
+Architecture discussion. To disentangle the effectiveness of the MP generator and discriminator, we train each individually with alternative counterparts (Table 2). With the same PointNet discriminator, the GraphCNN and TreeGAN generators perform worse than the simple FC generator for every metric on all three datasets. The physics-motivated MP generator on the other hand outperforms all on the gluon and top quark datasets, and significantly so on the jet-level $W _ { 1 }$ scores and the FPND. We note, however, that the MP generator is not a significant improvement over the other generators with an FC discriminator. Holding the generator fixed, the PointNet discriminator performs significantly better over the FC for all metrics. With the FC, GraphCNN, and TreeGAN generators, PointNet is also an improvement over the MP discriminator. With an MP generator, the MP discrimimator is more performant on jet-level $W _ { 1 }$ and FPND scores but, on the top quark dataset, degrades $W _ { 1 } ^ { \mathrm { P } }$ relative to PointNet.
+
+We learn from these three things: (1) a generator or discriminator architecture is only as effective as its counterpart—even though the MPGAN combination is the best overall, when paired with a network which is not able to learn complex substructure, or which breaks the permutation symmetry, neither the generator or discriminator is performant, (2) for high-fidelity jet feature reconstruction, both networks must be able to learn complex multi-particle correlations—however, this can come at the cost of low-level feature accuracy, and (3) MPGAN’s masking strategy is highly effective as both MP networks are improvements all around on light quark jets.
+
+Particle cloud evaluation metrics. We now discuss the merits of each evaluation metrics and provide suggestions for their use in future work. Fig. 4 shows correlation plots between chosen pairs of our evaluation metrics. As expected, we find W1-M and W1-EFP to be highly correlated, as they both measure learning of global jet features. For rigorous validation we suggest measuring both but for time-sensitive use-cases, such as quick evaluations during model training, W1-M should be sufficient. W1-M, FPND, and W1-P are all measuring different aspects of the generation and are relatively uncorrelated. We expect FPND overall to be the best and most discriminatory metric for evaluation, as it compares features found by a SOTA classifier to be statistically optimum for characterizing jets, while the W1 scores are valuable for their interpretability. Out of these, W1-M/W1-EFP are the most important from a physics-standpoint, as we generally characterize collisions by the high-level features of the output jets, rather than the individual particle features.
+
+MMD and coverage are both valuable for specifically evaluating the quality and diversity of samples respectively, however we see from Fig. 4 that they saturate after a certain point, after which FPND and $W _ { 1 }$ scores are necessary for stronger discrimination. We also note that in Table 2, models with low $W _ { 1 }$ scores relative to the baseline have the best coverage and MMD scores as well. This indicates that the $W _ { 1 }$ metrics are sensitive to both mode collapse (measured by coverage), which is expected as in terms of feature distributions mode collapse manifests as differing supports, to which the $W _ { 1 }$ distance is sensitive, as well as to individual sample quality (measured by MMD), which supports our claim that recovering jet feature distributions implies accurate learning of individual cloud structure. Together this suggests that low $W _ { 1 }$ scores are able validate sample quality and against mode collapse, and justifies our criteria that a practical ML simulation alternative have $W _ { 1 }$ scores close to the baselines in Table 2. In conclusion, for thorough validation of generated particle clouds, we recommend considering all three W-1 scores in conjunction with FPND, while MMD and coverage, being focused tests of these aspects of generation, may be useful for understanding failure modes during model development.
+
+![](images/92f879a528070bea4a2d1e4d4e65954f0e87a6f8dd0b9cb84a91f341143224dc.jpg)  
+Figure 4: Correlation plots between pairs of evaluation metrics, evaluated on 400 separate batches of 50,000 MPGAN generated top quark jets.
+
+# 6 Summary
+
+In this work, we publish JetNet: a novel particle cloud dataset to advance machine learning (ML) research in high energy physics (HEP), and provide a novel point-cloud-style dataset containing rich underlying physics for the ML community to experiment with. We apply existing state-of-the-art point cloud generative models to JetNet, and propose several physics- and computer-vision-inspired metrics to rigorously evaluate generated clouds. We find that existing models are not performant on a number of metrics, and fail to reproduce high-level jet features—arguably the most significant aspect for HEP. Our new message-passing generative adversarial network (MPGAN) model, designed to capture complex global structure and handle variable-sized clouds significantly improves performance in this area, as well as other metrics. We propose MPGAN as a new baseline model on JetNet and invite others to improve upon it.
+
+Impact With our JetNet dataset and library, we hope to lower the barrier to entry, improve reproducibility, and encourage development in HEP and ML. Particularly so in the area of simulation, where an accurate and fast ML particle cloud generator will have significant impact in (1) lowering the computational and energy cost of HEP research, as well as (2) increasing precision and sensitivity to new physics at the Large Hadron Collider and future colliders by providing more high-quality simulated data samples. One negative consequence of this, however, may be a loss of interpretability, and hence trustability, of the particle production generative model, which may ultimately increase uncertainties—though the metrics we propose should mitigate against this. More broadly, further advancements in the field of ML point cloud generation may result in fake visual data generation for proliferation of misinformation and impersonation/identity theft.
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 772369). R. K. was partially supported by an IRIS-HEP fellowship through the U.S. National Science Foundation (NSF) under Cooperative Agreement OAC-1836650, and by the LHC Physics Center at Fermi National Accelerator Laboratory, managed and operated by Fermi Research Alliance, LLC under Contract No. DE-AC02- 07CH11359 with the U.S. Department of Energy (DOE). J. D. is supported by the DOE, Office of Science, Office of High Energy Physics Early Career Research program under Award No. DESC0021187 and by the DOE, Office of Advanced Scientific Computing Research under Award No. DE-SC0021396 (FAIR4HEP). B. O and T. T are supported by grant 2018/25225-9, São Paulo Research Foundation (FAPESP). B. O was also partially supported by grants #2018/01398-1 and #2019/16401-0, São Paulo Research Foundation (FAPESP). J-R. V. is partially supported by the ERC under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 772369) and by the DOE, Office of Science, Office of High Energy Physics under Award No. DE-SC0011925, DE-SC0019227, and DE-AC02-07CH11359. D. G. is partially supported by the EU ICT-48 2020 project TAILOR (No. 952215). This work was performed using the Pacific Research Platform Nautilus HyperCluster supported by NSF awards CNS-1730158, ACI1540112, ACI-1541349, OAC-1826967, the University of California Office of the President, and the University of California San Diego’s California Institute for Telecommunications and Information Technology/Qualcomm Institute. Thanks to CENIC for the 100 Gpbs networks. Funding for cloud credits was supported by NSF Award #1904444 Internet2 supported E-CAS Exploring Clouds to Accelerate Science.
+
+# References
+
+[1] A. J. Larkoski, I. Moult, and B. Nachman, “Jet Substructure at the Large Hadron Collider: A Review of Recent Advances in Theory and Machine Learning”, Phys. Rept. 841 (2020) 1, doi:10.1016/j.physrep.2019.11.001, arXiv:1709.04464.   
+[2] L. de Oliveira, M. Paganini, and B. Nachman, “Learning particle physics by example: Location-aware generative adversarial networks for physics synthesis”, Comput. Softw. Big Sci. 1 (2017) 4, doi:10.1007/s41781-017-0004-6, arXiv:1701.05927.   
+[3] R. Kansal, “JETNET Library”. doi:10.5281/zenodo.5597893, https://github.com/jet-net/JetNet.   
+[4] R. K. Ellis, W. J. Stirling, and B. R. Webber, “QCD and collider physics”. Cambridge monographs on particle physics, nuclear physics, and cosmology. Cambridge University Press, Cambridge, 2003. Photography by S. Vascotto. doi:10.1017/CBO9780511628788.   
+[5] T. Sjöstrand et al., “An introduction to PYTHIA 8.2”, Comput. Phys. Commun. 191 (2015) 159, doi:10.1016/j.cpc.2015.01.024, arXiv:1410.3012.   
+[6] J. Bellm et al., “HERWIG 7.0/HERWIG $+ + 3 . 0$ release note”, Eur. Phys. J. C 76 (2016), no. 4, 196, doi:10.1140/epjc/s10052-016-4018-8, arXiv:1512.01178.   
+[7] B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, “Parton Fragmentation and String Dynamics”, Phys. Rept. 97 (1983) 31, doi:10.1016/0370-1573(83)90080-7.   
+[8] J. Cogan, M. Kagan, E. Strauss, and A. Schwarztman, “Jet-Images: Computer Vision Inspired Techniques for Jet Tagging”, JHEP 02 (2015) 118, doi:10.1007/JHEP02(2015)118, arXiv:1407.5675.   
+[9] L. de Oliveira et al., “Jet-images — deep learning edition”, JHEP 07 (2016) 069, doi:10.1007/JHEP07(2016)069, arXiv:1511.05190.   
+[10] H. Qu and L. Gouskos, “ParticleNet: Jet Tagging via Particle Clouds”, Phys. Rev. D 101 (2020), no. 5, 056019, doi:10.1103/PhysRevD.101.056019, arXiv:1902.08570.   
+[11] J. Shlomi, P. Battaglia, and J.-R. Vlimant, “Graph neural networks in particle physics”, Mach. Learn.: Sci. Technol. 2 (2021) 021001, doi:10.1088/2632-2153/abbf9a, arXiv:2007.13681.   
+[12] J. Duarte and J.-R. Vlimant, “Graph neural networks for particle tracking and reconstruction”, in Artificial Intelligence for High Energy Physics, P. Calafiura, D. Rousseau, and K. Terao, eds. World Scientific Publishing, 12, 2020. arXiv:2012.01249. Submitted to Int. J. Mod. Phys. A. doi:10.1142/12200.   
+[13] CMS Collaboration, “Identification of heavy, energetic, hadronically decaying particles using machine-learning techniques”, JINST 15 (2020) P06005, doi:10.1088/1748-0221/15/06/P06005, arXiv:2004.08262.   
+[14] A. X. Chang et al., “ShapeNet: An information-rich 3D model repository”, Technical Report, 2015. arXiv:1512.03012.   
+[15] R. Kansal et al., “JetNet”, 05, 2021. doi:10.5281/zenodo.4834876.   
+[16] M. Pierini, J. M. Duarte, N. Tran, and M. Freytsis, “hls4ml LHC jet dataset (30 particles)”, 01, 2020. doi:10.5281/zenodo.3601436.   
+[17] M. Paganini, L. de Oliveira, and B. Nachman, “CaloGAN $:$ Simulating 3D high energy particle showers in multilayer electromagnetic calorimeters with generative adversarial networks”, Phys. Rev. D 97 (2018), no. 1, 014021, doi:10.1103/PhysRevD.97.014021, arXiv:1712.10321.   
+[18] ATLAS Collaboration, “Fast simulation of the ATLAS calorimeter system with generative adversarial networks”, ATLAS Software Public Note ATL-SOFT-PUB-2020-006, 2020.   
+[19] M. Erdmann, J. Glombitza, and T. Quast, “Precise simulation of electromagnetic calorimeter showers using a Wasserstein generative adversarial network”, Comput. Softw. Big Sci. 3 (2019) 4, doi:10.1007/s41781-018-0019-7, arXiv:1807.01954.   
+[20] F. Carminati et al., “Generative adversarial networks for fast simulation”, J. Phys. Conf. Ser. 1525 (2020) 012064, doi:10.1088/1742-6596/1525/1/012064.   
+[21] Y. Lu, J. Collado, D. Whiteson, and P. Baldi, “Sparse autoregressive models for scalable generation of sparse images in particle physics”, Phys. Rev. D 103 (2021), no. 3, 036012, doi:10.1103/PhysRevD.103.036012, arXiv:2009.14017.   
+[22] S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift”, in Proceedings of the 32nd International Conference on Machine Learning, F. Bach and D. Blei, eds., volume 37, p. 448. PMLR, 2015. arXiv:1502.03167.   
+[23] C.-L. Liet al., “Point cloud GAN”, in 2019 ICLR Workshop Deep Generative Models for Highly Structured Data. 2018. arXiv:1810.05795.   
+[24] G. Yang et al., “PointFlow: 3D point cloud generation with continuous normalizing flows”, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), p. 4540. 2019. arXiv:1906.12320. doi:10.1109/ICCV.2019.00464.   
+[25] R. Klokov, E. Boyer, and J. Verbeek, “Discrete point flow networks for efficient point cloud generation”, in 2020 European Conference on Computer Vision (ECCV), A. Vedaldi, H. Bischof, T. Brox, and J.-M. Frahm, eds., p. 694. Springer International Publishing, Cham, 2020. arXiv:2007.10170.   
+[26] R. Cai et al., “Learning gradient fields for shape generation”, in 2020 European Conference on Computer Vision (ECCV), A. Vedaldi, H. Bischof, T. Brox, and J.-M. Frahm, eds., p. 364. Cham, 2020. arXiv:2008.06520.   
+[27] J. Köhler, L. Klein, and F. Noe, “Equivariant flows: Exact likelihood generative learning for symmetric densities”, in Proceedings of the 37th International Conference on Machine Learning, H. D. III and A. Singh, eds., volume 119 of Proceedings of Machine Learning Research, p. 5361. PMLR, 2020. arXiv:2006.02425.   
+[28] G. N. C. Simm, R. Pinsler, G. Csányi, and J. M. Hernández-Lobato, “Symmetry-aware actor-critic for 3d molecular design”, in International Conference on Learning Representations. 2021.   
+[29] N. Gebauer, M. Gastegger, and K. Schütt, “Symmetry-adapted generation of 3D point sets for the targeted discovery of molecules”, in Advances in Neural Information Processing Systems 32, H. Wallach et al., eds., p. 7566. Curran Associates, Inc., 2019. arXiv:1906.00957.   
+[30] P. Achlioptas, O. Diamanti, I. Mitliagkas, and L. Guibas, “Learning representations and generative models for 3D point clouds”, in Proceedings of the 35th International Conference on Machine Learning, J. Dy and A. Krause, eds., volume 80, p. 40. PMLR, 2018.   
+[31] D. Valsesia, G. Fracastoro, and E. Magli, “Learning localized generative models for 3D point clouds via graph convolution”, in International Conference on Learning Representations (ICLR) 2019. 2019.   
+[32] D. W. Shu, S. W. Park, and J. Kwon, “3D point cloud generative adversarial network based on tree structured graph convolutions”, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), p. 3858. 2019. arXiv:1905.06292. doi:10.1109/ICCV.2019.00396.   
+[33] C. R. Qi, H. Su, K. Mo, and L. J. Guibas, “PointNet: Deep learning on point sets for 3D classification and segmentation”, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). July, 2017.   
+[34] H. Wang et al., “Rethinking sampling in 3D point cloud generative adversarial networks”, 2020. arXiv:2006.07029.   
+[35] J. Gilmeret al., “Neural message passing for quantum chemistry”, in Proceedings of the 34th International Conference on Machine Learning, D. Precup and Y. W. Teh, eds., volume 70, p. 1263. PMLR, 2017. arXiv:1704.01212.   
+[36] CMS Collaboration, “Recent Developments in CMS Fast Simulation”, PoS ICHEP2016 (2016) 181, doi:10.22323/1.282.0181, arXiv:1701.03850.   
+[37] P. T. Komiske, E. M. Metodiev, and J. Thaler, “Energy flow polynomials: A complete linear basis for jet substructure”, JHEP 04 (2018) 013, doi:10.1007/JHEP04(2018)013, arXiv:1712.07124.   
+[38] A. Borji, “Pros and cons of GAN evaluation measures: New developments”, 2021. arXiv:2103.09396.   
+[39] M. Heuselet al., “GANs trained by a two time-scale update rule converge to a local nash equilibrium”, in Advances in Neural Information Processing Systems, I. Guyon et al., eds., volume 30, p. 6626. Curran Associates, Inc., 2017. arXiv:1706.08500.   
+[40] P. T. Komiske, E. M. Metodiev, and J. Thaler, “Metric Space of Collider Events”, Phys. Rev. Lett. 123 (2019), no. 4, 041801, doi:10.1103/PhysRevLett.123.041801, arXiv:1902.02346.   
+[41] Y. Wang et al., “Dynamic graph CNN for learning on point clouds”, ACM Trans. Graph. 38 (2019) doi:10.1145/3326362, arXiv:1801.07829.   
+[42] F. Monti et al., “Geometric deep learning on graphs and manifolds using mixture model CNNs”, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), p. 5425. 2017. arXiv:1611.08402. doi:10.1109/CVPR.2017.576.   
+[43] R. Kansal, “MPGAN Code”. doi:10.5281/zenodo.5598407, https://github.com/rkansal47/MPGAN/tree/neurips21.   
+[44] Pedro, Kevin, “Current and future performance of the cms simulation”, EPJ Web Conf. 214 (2019) 02036, doi:10.1051/epjconf/201921402036.   
+[45] C. Chen et al., “Data augmentation at the LHC through analysis-specific fast simulation with deep learning”, 2020. arXiv:2010.01835.   
+[46] K. Cranmer, J. Brehmer, and G. Louppe, “The frontier of simulation-based inference”, PNAS 117 (2020) 30055, doi:10.1073/pnas.1912789117, arXiv:1911.01429.   
+[47] G. Cowan, “Statistics”, Ch. 40 in Particle Data Group, P. A. Zyla et al., “Review of Particle Physics”, PTEP 2020 (2020) 083C01, doi:10.1093/ptep/ptaa104.   
+[48] CMS Collaboration, “Precise determination of the mass of the Higgs boson and tests of compatibility of its couplings with the standard model predictions using proton collisions at 7 and 8 TeV”, Eur. Phys. J. C 75 (2015) 212, doi:10.1140/epjc/s10052-015-3351-7, arXiv:1412.8662.   
+[49] R. J. Barlow and C. Beeston, “Fitting using finite Monte Carlo samples”, Comput. Phys. Commun. 77 (1993) 219, doi:10.1016/0010-4655(93)90005-W.   
+[50] J. Brehmer and K. Cranmer, “Simulation-based inference methods for particle physics”, in Artificial Intelligence for High Energy Physics, P. Calafiura, D. Rousseau, and K. Terao, eds. World Scientific Publishing, 10, 2020. arXiv:2010.06439. Submitted to Int. J. Mod. Phys. A. doi:10.1142/12200.   
+[51] J. Alwall et al., “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations”, JHEP 07 (2014) 079, doi:10.1007/JHEP07(2014)079, arXiv:1405.0301.   
+[52] R. D. Ball et al., “Parton distributions with LHC data”, Nucl. Phys. B 867 (2013) 244–289, doi:10.1016/j.nuclphysb.2012.10.003, arXiv:1207.1303.   
+[53] P. Skands, S. Carrazza, and J. Rojo, “Tuning PYTHIA 8.1: The Monash 2013 Tune”, Eur. Phys. J. C 74 (2014), no. 8, 3024, doi:10.1140/epjc/s10052-014-3024-y, arXiv:1404.5630.   
+[54] M. Cacciari, G. P. Salam, and G. Soyez, “The anti- $\cdot k _ { \mathrm { T } }$ jet clustering algorithm”, JHEP 04 (2008) 063, doi:10.1088/1126-6708/2008/04/063, arXiv:0802.1189.   
+[55] M. Cacciari, G. P. Salam, and G. Soyez, “FastJet user manual”, Eur. Phys. J. C 72 (2012) 1896, doi:10.1140/epjc/s10052-012-1896-2, arXiv:1111.6097.   
+[56] M. Cacciari and G. P. Salam, “Dispelling the $n ^ { 3 }$ myth for the $k _ { \mathrm { T } }$ jet-finder”, Phys. Lett. B 641 (2006) 57, doi:10.1016/j.physletb.2006.08.037, arXiv:hep-ph/0512210.   
+[57] X. Mao et al., “Multi-class generative adversarial networks with the L2 loss function”, 2016. arXiv:1611.04076.   
+[58] N. Srivastavaet al., “Dropout: A simple way to prevent neural networks from overfitting”, J. Mach. Learn. Res. 15 (2014) 1929.   
+[59] I. Gulrajaniet al., “Improved training of Wasserstein GANs”, in Advances in Neural Information Processing Systems, I. Guyon et al., eds., volume 30, p. 5767. Curran Associates, Inc., 2017. arXiv:1704.00028.   
+[60] T. Miyato, T. Kataoka, M. Koyama, and Y. Yoshida, “Spectral normalization for generative adversarial networks”, in 6th International Conference on Learning Representations. 2018. arXiv:1802.05957.   
+[61] F. Schäfer, H. Zheng, and A. Anandkumar, “Implicit competitive regularization in GANs”, 2019. arXiv:1910.05852.   
+[62] T. Karraset al., “Training generative adversarial networks with limited data”, in Advances in Neural Information Processing Systems, H. Larochelle et al., eds., volume 33, p. 12104. Curran Associates, Inc., 2020. arXiv:2006.06676.   
+[63] N.-T. Tran et al., “On data augmentation for GAN training”, IEEE Trans. Image Process. 30 (2021) 1882, doi:10.1109/tip.2021.3049346, arXiv:2006.05338.   
+[64] Z. Zhao et al., “Image augmentations for GAN training”, 2020. arXiv:2006.02595.   
+[65] M. Arjovsky, S. Chintala, and L. Bottou, “Wasserstein GAN”, 2017. arXiv:1701.07875.

parse/train/iorEu783qJ5/iorEu783qJ5_content_list.json

@@ -0,0 +1,877 @@
+[
+    {
+        "type": "text",
+        "text": "Particle Cloud Generation with Message Passing Generative Adversarial Networks ",
+        "text_level": 1,
+        "bbox": [
+            207,
+            122,
+            794,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Raghav Kansal, Javier Duarte, Hao Su University of California San Diego La Jolla, CA 92093, USA ",
+        "bbox": [
+            214,
+            222,
+            488,
+            270
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Breno Orzari, Thiago Tomei Universidade Estadual Paulista São Paulo/SP - CEP 01049-010, Brazil ",
+        "bbox": [
+            529,
+            222,
+            787,
+            270
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Maurizio Pierini, Mary Touranakou⇤ European Organization for Nuclear Research (CERN) CH-1211 Geneva 23, Switzerland ",
+        "bbox": [
+            194,
+            290,
+            549,
+            338
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jean-Roch Vlimant California Institute of Technology Pasadena, CA 91125, USA ",
+        "bbox": [
+            576,
+            291,
+            802,
+            339
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dimitrios Gunopulos National and Kapodistrian University of Athens Athens 15772, Greece ",
+        "bbox": [
+            339,
+            361,
+            656,
+            407
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            444,
+            535,
+            462
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In high energy physics (HEP), jets are collections of correlated particles produced ubiquitously in particle collisions such as those at the CERN Large Hadron Collider (LHC). Machine learning (ML)-based generative models, such as generative adversarial networks (GANs), have the potential to significantly accelerate LHC jet simulations. However, despite jets having a natural representation as a set of particles in momentum-space, a.k.a. a particle cloud, there exist no generative models applied to such a dataset. In this work, we introduce a new particle cloud dataset (JetNet), and apply to it existing point cloud GANs. Results are evaluated using (1) 1-Wasserstein distances between high- and low-level feature distributions, (2) a newly developed Fréchet ParticleNet Distance, and (3) the coverage and (4) minimum matching distance metrics. Existing GANs are found to be inadequate for physics applications, hence we develop a new message passing GAN (MPGAN), which outperforms existing point cloud GANs on virtually every metric and shows promise for use in HEP. We propose JetNet as a novel point-cloud-style dataset for the ML community to experiment with, and set MPGAN as a benchmark to improve upon for future generative models. Additionally, to facilitate research and improve accessibility and reproducibility in this area, we release the open-source JETNET Python package with interfaces for particle cloud datasets, implementations for evaluation and loss metrics, and more tools for ML in HEP development. ",
+        "bbox": [
+            232,
+            477,
+            766,
+            738
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            763,
+            310,
+            781
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Over the past decade, machine learning (ML) has become the de facto way to analyze jets, collimated high-energy sprays of particles [1] produced at the CERN Large Hadron Collider (LHC). To apply ML to jets, the most natural representation is a particle cloud, a variable-sized set of points in momentum space, whose radiation pattern contains rich information about the underlying physics known as quantum chromodynamics (QCD). A fundamental question is whether ML algorithms can model this underlying physics and successfully reproduce the rich high- and low-level structure in jets. ",
+        "bbox": [
+            174,
+            795,
+            825,
+            878
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Answering this question affirmatively has important practical applications. At the LHC, large simulated data samples of collision events2 are generated using Monte Carlo (MC) methods in order to translate theoretical predictions into observable distributions, and ultimately perform physics analyses3. These samples, numbering in the billions of events, require computationally expensive modeling of the interaction of particles traversing the detector material. Recently developed generative frameworks in ML such as generative adversarial networks (GANs), if accurate enough, can be used to accelerate this simulation by potentially five orders of magnitude [2]. ",
+        "bbox": [
+            174,
+            92,
+            825,
+            188
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we advocate for a benchmark jet dataset (JetNet) and propose several physics- and computer-vision-inspired metrics with which the ML community can improve and evaluate generative models in high energy physics (HEP). To facilitate and encourage research in this area, as well as to make such research more accessible and reproducible, we release interfaces for public particle cloud datasets such as JetNet, implementations for our proposed metrics, and various tools for ML in HEP development in the JETNET library [3]. We next apply existing point cloud GANs on JetNet and find the results to be inadequate for physics applications. Finally, we develop our own message passing GAN (MPGAN), which dramatically improves results on virtually every metric, and propose it as a benchmark on JetNet. ",
+        "bbox": [
+            174,
+            194,
+            825,
+            319
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Jets ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            342,
+            238,
+            358
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "High-energy proton-proton collisions at the LHC produce elementary particles like quarks and gluons, which cannot be isolated due to the QCD property of color confinement [4]. These particles continuously radiate or “split” into a set of particles, known as a parton shower. Eventually they cool to an energy at which they undergo the process of hadronization, where the fundamental particles combine to form more stable hadrons, such as pions and protons. The final set of collimated hadrons produced after such a process is referred to as a jet. ",
+        "bbox": [
+            174,
+            375,
+            825,
+            458
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The task of simulating a single jet can be algorithmically defined as inputting an initial particle, which produces the jet, and outputting the final set of particles a.k.a. the jet constituents. Typically in HEP the parton shower and hadronization are steps that are simulated sequentially using MC event generators such as PYTHIA [5] or HERWIG [6]. Simulating either process exactly is not possible because of the complex underlying physics (QCD), and instead these event generators fit simplified physics-inspired stochastic models, such as the Lund string model for hadronization [7], to existing data using MC methods. The present work can be seen as an extension of this idea, using a simpler, ML-based model, also fitted to data, for generating the jet in one shot, where we are effectively trading the interpretability of the MC methods for the speed of GPU-accelerated ML generators. ",
+        "bbox": [
+            174,
+            464,
+            825,
+            588
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Representations. As common for collider physics, we use a Cartesian coordinate system with the $z$ axis oriented along the beam axis, the $x$ axis on the horizontal plane, and the $y$ axis oriented upward. The $x$ and $y$ axes define the transverse plane, while the $z$ axis identifies the longitudinal direction. The azimuthal angle $\\phi$ is computed with respect to the $x$ axis. The polar angle $\\theta$ is used to compute the pseudorapidity $\\eta = - \\log ( \\tan ( \\theta / 2 ) )$ . The transverse momentum $( p _ { \\mathrm { T } } )$ is the projection of the particle momentum on the $( x , y )$ plane. As is customary, we transform the particle momenta from Cartesian coordinates $\\left( p _ { x } , p _ { y } , p _ { z } \\right)$ to longitudinal-boost-invariant pseudo-angular coordinates $( p _ { \\mathrm { T } } , \\eta , \\phi )$ , as shown in Fig. 1. ",
+        "bbox": [
+            173,
+            606,
+            825,
+            718
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the context of ML, jets can be represented in multiple ways. One popular representation is as images [8, 9], created by projecting each jet’s particle constituents onto a discretized angular $\\eta { - } \\phi$ plane, and taking the intensity of each “pixel” in this grid to be a monotonically increasing function of the corresponding particle $p _ { \\mathrm { T } }$ . These tend to be extremely sparse, with typically fewer than $10 \\%$ of pixels nonempty [10], and the discretization process can furthemore lower the resolution. ",
+        "bbox": [
+            174,
+            723,
+            825,
+            792
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Two more spatially efficient representations are as ordered lists or unordered sets [11, 12] of the jet constituents and their features. The difficulty with the former is that there is no particular preferred ordering of the particles—one would have to impose an arbitrary ordering such as by transverse momentum [13]. The more natural representation is the unordered set of particles in momentum space, which we refer to as a “particle cloud.” This is in analogy to point cloud representations of 3D objects in position-space prevalent in computer vision created, for example, by sampling from 3D ShapeNet models [14]. ",
+        "bbox": [
+            174,
+            799,
+            825,
+            868
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/84898c4b21c9c3390cd9178cd2c4e55c4ea0b64586f4d8d3d28bd30f3b2d3275.jpg",
+        "image_caption": [
+            "Figure 1: The collider physics coordinate system defining $( p _ { \\mathrm { T } } , \\eta , \\phi )$ (left). The three jet classes in our dataset (right). Gluon (g) and light quark (q) jets have simple topologies, with q jets generally containing fewer particles. Top quark (t) jets have a complex three-pronged structure. Shown also are the relative angular coordinates $\\bar { \\boldsymbol { \\eta } } ^ { \\mathrm { r e l } }$ h→and $\\dot { \\phi } ^ { \\mathrm { r e l } }$ t→Wb→q, measured from the jet axis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            92,
+            813,
+            210
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            305,
+            821,
+            333
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Apart from how the samples are produced, significant differences between jets and ShapeNet-based point clouds are that, firstly, jets have physically meaningful low- and high-level features such as particle momentum, total mass of the jet, the number of sub-jets, and $n$ -particle energy correlations. These physical observables are how we characterize jets, and hence are important to reproduce correctly for physics analysis applications. Secondly, unlike the conditional distributions of points given a particular ShapeNet object, which are identical and independent, particle distributions within jets are highly correlated, as the particles each originate from a single source. The independence of their constituents also means that ShapeNet-sampled point clouds can be chosen to be of a fixed cardinality, whereas this is not possible for jets, which inherently contain varying numbers of particles due to the stochastic nature of particle production. ",
+        "bbox": [
+            173,
+            339,
+            825,
+            478
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "JetNet. We publish JetNet [15] under the CC-BY 4.0 license, to facilitate and advance ML research in HEP, and to offer a new point-cloud-style dataset to experiment with. Derived from Ref. $[ 1 6 ] ^ { 4 }$ , it consists of simulated particle jets with transverse momenta $p _ { \\mathrm { T } } ^ { \\mathrm { j e t } } \\approx 1 \\mathrm { T e V }$ , originating from gluons, light quarks, and top quarks produced in $1 3 \\mathrm { T e V }$ proton-proton collisions in a simplified detector. Technical details of the generation process are given in App. B. We limit the number of constituents to the 30 highest $p _ { \\mathrm { T } }$ particles per jet, allowing for jets with potentially fewer than 30 by zeropadding. For each particle we provide the following four features: the relative angular coordinates $\\bar { \\eta } ^ { \\mathrm { r e l } } = \\bar { \\eta } ^ { \\mathrm { p a r t i c l e } } - \\bar { \\eta } ^ { \\mathrm { j e t } }$ and $\\phi ^ { \\mathrm { r e l } } { \\bar { = } } \\phi ^ { \\mathrm { p a r t i c l e } } - \\phi ^ { \\mathrm { j e t } }$ (mod $2 \\pi$ ), relative transverse momentum $p _ { \\mathrm { T } } ^ { \\mathrm { r e l } } =$ $p _ { \\mathrm { T } } ^ { \\mathrm { p a r t i c l e } } / p _ { \\mathrm { T } } ^ { \\mathrm { j e t } }$ , and a binary mask feature classifying the particle as genuine or zero-padded. ",
+        "bbox": [
+            173,
+            492,
+            825,
+            623
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We choose three jet classes, depicted in Fig. 1, to individually target the unique and challenging properties of jets. Gluons provide a useful baseline test, as they typically radiate into a large number of particles before hadronization, largely avoiding the variable-sized cloud issue—at least with a 30 particle maximum, and have a relatively simple topology. Light quarks share the simple topology, but produce fewer final-state particles, resulting in a larger fraction of zero-padded particles in the dataset. They allow evaluation of a model’s ability to handle variable-sized clouds. Finally, top quarks decay into three lighter quarks through an intermediate particle, the W boson, which each may produce their own sub-jets, leading to a complex two- or three-pronged topology—depending on whether the jet clustering algorithm captures all three or just two of these sub-jets. This results in bimodal jet feature distributions (one peak corresponding to fully merged top quark jets and the other to semi-merged, as seen in Fig. 3). Thus, top quark jets test models’ ability to learn the rich global structure and clustering history of a particle cloud. ",
+        "bbox": [
+            173,
+            628,
+            825,
+            795
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            814,
+            321,
+            830
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Generative models in HEP. Past work in this area has exclusively used image-based representations for HEP data. One benefit of this is the ability to employ convolutional neural network (CNN) based generative models, which have been highly successful on computer vision tasks. ",
+        "bbox": [
+            176,
+            845,
+            826,
+            887
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Refs. [2, 17–20], for example, build upon CNN-based GANs, and Ref. [21] uses an auto-regressive model, to output jet- and detector-data-images. ",
+        "bbox": [
+            173,
+            90,
+            823,
+            119
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In addition to the issues with such representations outlined in Sec. 2, the high sparsity of the images can lead to training difficulties in GANs, and the irregular geometry of the data — a single LHC detector can typically have multiple sections with differing pixel sizes and shapes — poses a challenge for CNN GANs which output uniform matrices. While these can be mitigated to an extent with techniques such as batch normalization [22] and using larger/more regular pixels [18], our approach avoids both issues by generating particle-cloud-representations of the data, as these are inherently sparse data structures and are completely flexible to the underlying geometry. ",
+        "bbox": [
+            174,
+            126,
+            825,
+            223
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "GANs for point clouds. There are several published generative models in this area, however the majority exploit inductive biases specific to their respective datasets, such as ShapeNet-based [23–26] and molecular [27–29] point clouds, which are not appropriate for jets. A more detailed discussion, including some experimental results, can be found in App. C. ",
+        "bbox": [
+            174,
+            237,
+            825,
+            292
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "There do exist some more general-purpose GAN models, namely r-GAN [30], GraphCNN-GAN [31], and TreeGAN [32], and we test these on JetNet. r-GAN uses a fully-connected (FC) network, GraphCNN-GAN uses graph convolutions based on dynamic $k$ -nn graphs in intermediate feature spaces, and TreeGAN iteratively up-samples the graphs with information passing from ancestor to descendant nodes. In terms of discriminators, past work has used either a FC or a PointNet [33]-style network. Ref. [34] is the first work to study point cloud discriminator design in detail and finds amongst a number of PointNet and graph convolutional models that PointNet-Mix, which uses both max- and average-pooled features, is the most performant. ",
+        "bbox": [
+            174,
+            299,
+            825,
+            410
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We apply the three aforementioned generators and FC and PointNet-Mix discriminators as baselines to our dataset, but find jet structure is not adequately reproduced. GraphCNN’s local convolutions make learning global structure difficult, and while the TreeGAN and FC generator $^ +$ PointNet discriminator combinations are improvements, they are not able to learn multi-particle correlations, particularly for the complex top quark jets, nor deal with the variable-sized light quark jets to the extent necessary for physics applications. ",
+        "bbox": [
+            174,
+            416,
+            825,
+            500
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Message Passing Neural Networks. We attempt to overcome limitations of existing GANs by designing a novel generator and discriminator which can learn such correlations and handle variablesized particle clouds. Both networks build upon the generic message-passing neural network (MPNN) [35] framework with physics-conscious design choices, and collectively we refer to them as message-passing GAN (MPGAN). We find MPGAN outperforms existing models on virtually all evaluation metrics. ",
+        "bbox": [
+            174,
+            513,
+            825,
+            597
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Evaluating generative models. ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            613,
+            421,
+            627
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Evaluating generative models is a difficult task, however there has been extensive work in this area in both the physics and computer-vision communities. ",
+        "bbox": [
+            174,
+            638,
+            823,
+            666
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Physics-inspired metrics. An accurate jet simulation algorithm should reproduce both low-level and high-level features (such as those described in Sec. 2), hence a standard method of validating generative models, which we employ, is to compare the distributions of such features between the real and generated samples5 [2, 17–20, 36]. ",
+        "bbox": [
+            174,
+            681,
+            825,
+            737
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For application in HEP, a generative model needs to produce jets with physical features indistinguishable from real. Therefore, we propose the validation criteria that differences between real and generated sample features may not exceed those between sets of randomly chosen real samples. To verify this, we use bootstrapping to compare between random samples of only real jets as a baseline. ",
+        "bbox": [
+            174,
+            743,
+            825,
+            799
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A practically useful set of features to validate against are the so-called “energy-flow polynomials” (EFPs) [37], which are a set of multi-particle correlation functions. Importantly, the set of all EFPs forms a linear basis for all useful jet-level features6. Therefore, we claim that if we observe all EFP distributions to be reproduced with high fidelity and to match the above criteria, we can conclude with strong confidence that our model is outputting accurate particle clouds. ",
+        "bbox": [
+            176,
+            804,
+            825,
+            875
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/5fece647d1585c77642972c14b6d56ef8139794d5fd92b6e0f81e0f4bbc17a39.jpg",
+        "image_caption": [
+            "Figure 2: Top: The MP generator uses message passing to generate a particle cloud. In blue is the initial latent vector and FC layer part of the MP-LFC variant. Bottom: The MP discriminator uses message passing to classify an input particle cloud as real or generated. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            90,
+            826,
+            411
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Computer-vision-inspired metrics A popular metric for evaluating images which has shown to be sensitive to output quality and mode-collapse, though it has its limitations [38], is the Fréchet Inception Distance [39] (FID). FID is defined as the Fréchet distance between Gaussian distributions fitted to the activations of a fully-connected layer of the Inception-v3 image classifier in response to real and generated samples. We develop a particle-cloud-analogue of this metric, which we call Fréchet ParticleNet Distance (FPND), using the state-of-the-art (SOTA) ParticleNet graph convolutional jet classifier [10] in lieu of the Inception network. We note that the FPND and comparing distributions as above is conceptually equivalent, except here instead of physically meaningful and easily interpretable features, we are comparing those found to be statistically optimum for distinguishing jets. ",
+        "bbox": [
+            174,
+            491,
+            825,
+            616
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Two common metrics for evaluating point cloud generators are coverage (COV) and minimum matching distance (MMD) [30]. Both involve finding the closest point cloud in a sample $X$ to each cloud in another sample $Y$ , based on a metric such as the Chamfer distance or the earth mover’s distance. Coverage is defined as the fraction of samples in $X$ which were matched to one in $Y$ , measuring thus the diversity of the samples in $Y$ relative to $X$ , and MMD is the average distance between matched samples, measuring the quality of samples. We use both, and due to drawbacks of the Chamfer distance pointed out in Ref. [30], for our distance metric choose only the analogue of the earth mover’s distance for particle clouds a.k.a. the energy mover’s distance (EMD) [40]. We discuss the effectiveness and complementarity of all four metrics in evaluating clouds in Sec. 5. ",
+        "bbox": [
+            174,
+            622,
+            825,
+            747
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 MPGAN Architecture ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            766,
+            390,
+            784
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We describe now the architecture of our MPGAN model (Fig. 2), noting particle cloud-motivated aspects compared to its r-GAN and GraphCNN-GAN predecessors. ",
+        "bbox": [
+            174,
+            797,
+            823,
+            827
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Message passing. Jets originate from a single source particle decaying and hadronizing, hence they end up with important high-level jet features and a rich global structure, known as the jet substructure [1], stemming from the input particle. Indeed any high-level feature useful for analyzing jets, such as jet mass or multi-particle correlations, is necessarily global [37]. Because of this, while past work in learning on point clouds [10, 41, 42], including GraphCNN-GAN, has used a locally connected graph structure and convolutions for message passing, we choose a fully connected graph, equally weighting messages from all particles in the clouds. Rather than subtracting particle features for messages between particles, useful in graph convolutions to capture local differences within a neighborhood, the respective features are concatenated to preserve the global structure (the difference between particle features is also only physically meaningful if they are in the 4-vector representation of the Lorentz group). During the update step in the message passing we find it empirically beneficial to incorporate a residual connection to previous particle features. ",
+        "bbox": [
+            174,
+            842,
+            823,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            825,
+            189
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The operation can be described as follows. For an $N$ -particle cloud $J ^ { t } = \\{ p _ { 1 } ^ { t } , \\cdot \\cdot \\cdot , p _ { N } ^ { t } \\}$ after $t$ iterations of message passing, with $t = 0$ corresponding to the original input cloud, each particle $p _ { i } ^ { t }$ is represented by features $\\mathbf { h } _ { i } ^ { t }$ . One iteration of message passing is then defined as ",
+        "bbox": [
+            174,
+            194,
+            825,
+            237
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/38ba265e7663737061a739a39f897348c278fb876975af1bc653f43f7e5a4a5a.jpg",
+        "text": "$$\n\\begin{array} { r l } { { } } & { { \\mathbf { m } _ { i j } ^ { t + 1 } = f _ { e } ^ { t + 1 } ( \\mathbf { h } _ { i } ^ { t } \\oplus \\mathbf { h } _ { j } ^ { t } ) , } } \\\\ { { } } & { { \\mathbf { h } _ { i } ^ { t + 1 } = f _ { n } ^ { t + 1 } ( \\mathbf { h } _ { i } ^ { t } \\oplus \\displaystyle \\sum _ { j \\in J } \\mathbf { m } _ { i j } ^ { t + 1 } ) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            243,
+            606,
+            303
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where particl $\\mathbf { m } _ { i j } ^ { t + 1 }$ id mesand vector sent from particle are arbitrary functions w $j$ to particle ich, in our $i$ , $\\mathbf { h } _ { i } ^ { t + 1 }$ are the updated features of implemented as multilayer $i$ $f _ { e } ^ { t + 1 }$ $f _ { n } ^ { t + 1 }$   \nperceptrons (MLPs) with 3 FC layers. ",
+        "bbox": [
+            174,
+            310,
+            825,
+            356
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Generator. We test two initializations of a particle cloud for the MPGAN generator: (1) directly initializing the cloud with $N$ particles with $L$ randomly sampled features, which we refer to as the MP generator, and (2) inputting a single $Z$ -dimensional latent noise vector and transforming it via an FC layer into an $N \\times L$ -dimensional matrix, which we refer to as the MP-Latent-FC (MP-LFC) generator. The MP-LFC uses a latent space which can intuitively be understood as representing the initial source particle’s features along with parameters to capture the stochasticity of the jet production process. Due to the complex nature of this process, however, we posit that this global, flattened latent space cannot capture the full phase space of individual particle features. Hence, we introduce the MP generator, which samples noise directly per particle, and find that it outperforms MP-LFC (Table 2). ",
+        "bbox": [
+            173,
+            371,
+            825,
+            496
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Discriminator. We find the MP generator, in conjunction with a PointNet discriminator, to be a significant improvement on every metric compared to FC and GraphCNN generators. However, the jet-level features are not yet reproduced to a high enough accuracy (Sec. 5). While PointNet is able to capture global structural information, it can miss the complex interparticle correlations in real particle clouds. We find we can overcome this limitation by incorporating message passing in the discriminator as well as in the generator. Concretely, our MP discriminator receives the real or generated cloud and applies MP layers to produce intermediate features for each particle, which are then aggregated via a feature-wise average-pooling operation and passed through an FC layer to output the final scalar feature. We choose 2 MP layers for both networks. ",
+        "bbox": [
+            174,
+            511,
+            825,
+            636
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Variable-sized clouds. In order to handle clouds with varying numbers of particles, as typical of jets, we introduce an additional binary “masking” particle feature classifying the particle as genuine or zero-padded. Particles in the zero-padded class are ignored entirely in the message passing and pooling operations. The MP generator adds mask features to the initial particle cloud, using an additional input of the size of the jet $N$ , sampled from the real distribution, before the message passing layers based on sorting in particle feature space. Ablation studies with alternative (as well as no) masking strategies are discussed in App. E. ",
+        "bbox": [
+            173,
+            651,
+            825,
+            750
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            768,
+            312,
+            786
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Evaluation. We use four techniques discussed in Sec. 3.1 for evaluating and comparing models. Distributions of physical particle and jet features are compared visually and quantitatively using the Wasserstein-1 $( W _ { 1 } )$ distance between them. For ease of evaluation, we report (1) the average scores of the three particle features $( W _ { 1 } ^ { \\mathrm { P } } ) \\eta ^ { \\mathrm { r e l } }$ , $\\phi ^ { \\mathrm { r e l } }$ , and $p _ { \\mathrm { T } } ^ { \\mathrm { r e l } }$ , (2) the jet mass $( W _ { 1 } ^ { \\mathrm { M } } )$ , and (3) the average of a subset of the $\\mathrm { E F P s } ^ { 7 } ( W _ { 1 } ^ { \\mathrm { E F P } } )$ , which together provide a holistic picture of the low- and high-level aspects of a jet. The $W _ { 1 }$ distances are calculated for each feature between random samples of 10,000 real and generated jets, and averaged over 5 batches. Baseline $W _ { 1 }$ distances are calculated between two sets of randomly sampled real jets with 10,000 samples each, and are listed for each feature in Table 1. The real samples are split 70/30 for training/evaluation. We train ParticleNet for classification on our dataset to develop the FPND metric. FPND is calculated between 50,000 random real and generated samples, based on the activations of the first FC layer in our trained model8. Coverage and MMD are calculated between 100 real and 100 generated samples, and averaged over 10 such batches. Implementations for all metrics are provided in the JETNET package [3]. ",
+        "bbox": [
+            174,
+            800,
+            825,
+            886
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/c2247d42766906b95ee692494783ca41b8b194a8b8e989d54df76510d8898e87.jpg",
+        "table_caption": [
+            "Table 1: $W _ { 1 }$ distances between real jet mass $( W _ { 1 } ^ { \\mathrm { M } } )$ , averaged particle features $( W _ { 1 } ^ { \\mathrm { P } } )$ , and averaged jet EFPs $( W _ { 1 } ^ { \\mathrm { E F P } } )$ distributions calculated as a baseline, for three classes of jets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Jet class</td><td>WM (x10-3)</td><td>WP (×10-3)</td><td>WEFP (×10-5)</td></tr><tr><td>Gluon</td><td>0.7 ± 0.2</td><td>0.44± 0.09</td><td>0.62 ± 0.07</td></tr><tr><td>Light quark</td><td>0.5 ± 0.1</td><td>0.5 ± 0.1</td><td>0.46 ± 0.04</td></tr><tr><td>Top quark</td><td>0.51 ± 0.07</td><td>0.55 ± 0.07</td><td>1.1 ± 0.1</td></tr></table>",
+        "bbox": [
+            276,
+            127,
+            720,
+            209
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            234,
+            826,
+            333
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/51b648485cf27e970a00bad6019c1f8ab5c89a8359435edd729031e5c52bafd2.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of real and generated distributions for a subset of jet and particle features. We use the best performing model for each of the FC, GraphCNN, TreeGAN, and MP generators, as per Table 2. Top: gluon jet features, Middle: light quark jets, Bottom: top quark jets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            354,
+            825,
+            685
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results. On each of JetNet’s three classes, we test r-GAN’s FC, GraphCNN, and TreeGAN generators with rGAN’s FC and the PointNet-Mix discriminators, and compare them to MPGAN’s MP generator and discriminator models, including both MP and MP-LFC generator variations. Training and implementation details for each can be found in App. D, and all code in Ref. [43]. ",
+        "bbox": [
+            174,
+            755,
+            825,
+            810
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We choose model parameters which, during training, yield the lowest $W _ { 1 } ^ { \\mathrm { M } }$ score. This is because (1) $W _ { 1 }$ scores between physical features are more relevant for physics applications than the other three metrics, and (2) qualitatively we find it be a better discriminator of model quality than particle features or EFP scores. Table 2 lists the scores for each model and class, and Fig. 3 shows plots of selected feature distributions of real and generated jets, for the best performing FC, GraphCNN, TreeGAN, and MP generators. We also provide in App. F discretized images in the angular-coordinates-plane a.k.a “jet images”, however, we note that it is in general not easy to visually evaluate the quality of individual particle clouds, hence we focus on metrics and visualizations aggregated over batches of clouds. Overall we find that MPGAN is a significant improvement over the best FC, GraphCNN, and TreeGAN models, particularly for top and light quark jets. This is evident both visually and quantitatively in every metric, especially jet $W _ { 1 } s$ and FPND, with the exception of $W _ { 1 } ^ { \\mathrm { P } }$ where only the FC generator and PointNet discriminator $\\mathrm { F C } +$ PointNet) combination is more performant. ",
+        "bbox": [
+            174,
+            816,
+            825,
+            887
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/efe5b0d93563065f825c397c40a4394db18b535db92697eb2ac071ddae4a11a9.jpg",
+        "table_caption": [
+            "Table 2: Six evaluation scores on different generator and discriminator combinations. Lower is better for all metrics except COV. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Jet class</td><td rowspan=\"2\">Generator</td><td rowspan=\"2\">Discriminator</td><td rowspan=\"2\">WM (x10-3)</td><td rowspan=\"2\">WP (×10-3)</td><td rowspan=\"2\">WEFP (×10-5)</td><td rowspan=\"2\">FPND</td><td rowspan=\"2\">MMD</td></tr><tr><td>COV ↑</td></tr><tr><td rowspan=\"12\">Gluon</td><td>FC</td><td>FC</td><td>18.3± 0.2</td><td>9.6± 0.4</td><td>8.5±0.5</td><td>176</td><td>0.24</td><td>0.045</td></tr><tr><td>GraphCNN</td><td>FC</td><td>2.6± 0.2</td><td>9.6 ±0.3</td><td>12±8</td><td>61</td><td>0.39</td><td>0.046</td></tr><tr><td>TreeGAN</td><td>FC</td><td>41.9 ± 0.3</td><td>69.3 ± 0.3</td><td>14.2 ± 0.8</td><td>355</td><td>0.19</td><td>0.130</td></tr><tr><td>FC</td><td>PointNet</td><td>1.3 ± 0.4</td><td>1.3± 0.2</td><td>1.5 ± 0.9</td><td>5.0</td><td>0.49</td><td>0.039</td></tr><tr><td>GraphCNN</td><td>PointNet</td><td>1.9 ±0.2</td><td>16±6</td><td>200 ±1000</td><td>7k</td><td>0.46</td><td>0.040</td></tr><tr><td>TreeGAN</td><td>PointNet</td><td>1.7 ± 0.1</td><td>4.0± 0.4</td><td>4±1</td><td>84</td><td>0.37</td><td>0.042</td></tr><tr><td>MP</td><td>MP</td><td>0.7± 0.2</td><td>0.9 ± 0.3</td><td>0.7±0.2</td><td>0.12</td><td>0.56</td><td>0.037</td></tr><tr><td>MP-LFC</td><td>MP</td><td>0.69 ± 0.07</td><td>1.8± 0.2</td><td>0.9 ±0.6</td><td>0.20</td><td>0.54</td><td>0.037</td></tr><tr><td>FC</td><td>MP</td><td>4.3 ± 0.3</td><td>21.1 ±0.2</td><td>9±1</td><td>368</td><td>0.11</td><td>0.085</td></tr><tr><td>GraphCNN</td><td>MP</td><td>2.5 ± 0.1</td><td>9.8± 0.2</td><td>13±8</td><td>61</td><td>0.38</td><td>0.048</td></tr><tr><td>TreeGAN</td><td>MP</td><td>2.4± 0.2</td><td>12±7</td><td>18±9</td><td>69</td><td>0.34</td><td>0.048</td></tr><tr><td>MP</td><td>FC</td><td>1.2 ± 0.2</td><td>3.7 ± 0.5</td><td>1.6 ± 0.8</td><td>39</td><td>0.44</td><td>0.040</td></tr><tr><td>MP</td><td>PointNet</td><td>1.3± 0.4</td><td>1.2 ± 0.4</td><td>4±2</td><td>18</td><td>0.53</td><td>0.036</td></tr><tr><td rowspan=\"14\">Light quark</td><td>FC</td><td>FC</td><td>6.0±0.2</td><td>16.3 ± 0.9</td><td>3.9 ±0.6</td><td>395</td><td>0.18</td><td>0.053</td></tr><tr><td>GraphCNN</td><td>FC</td><td>3.5± 0.2</td><td>15.1 ± 0.4</td><td>10±50</td><td>100</td><td>0.25</td><td>0.038</td></tr><tr><td>TreeGAN</td><td>FC</td><td>31.5 ± 0.3</td><td>22.3±0.4</td><td>9.3 ± 0.4</td><td>176</td><td>0.06</td><td>0.055</td></tr><tr><td>FC</td><td>PointNet</td><td>3.1 ± 0.2</td><td>4.5± 0.4</td><td>2.3 ± 0.6</td><td>17</td><td>0.37</td><td>0.028</td></tr><tr><td>GraphCNN</td><td>PointNet</td><td>4±1</td><td>5.2±0.5</td><td>50k±100k</td><td>316</td><td>0.37</td><td>0.031</td></tr><tr><td>TreeGAN</td><td>PointNet</td><td>10.1 ± 0.1</td><td>5.7±0.5</td><td>4.1 ± 0.3</td><td>11</td><td>0.47</td><td>0.031</td></tr><tr><td>MP</td><td>MP</td><td>0.6±0.2</td><td>4.9 ± 0.5</td><td>0.7± 0.4</td><td>0.35</td><td>0.50</td><td>0.026</td></tr><tr><td>MP-LFC</td><td>MP</td><td>0.7±0.2</td><td>2.6 ± 0.4</td><td>0.9 ± 0.9</td><td>0.08</td><td>0.52</td><td>0.024</td></tr><tr><td>FC</td><td>MP</td><td>6.3± 0.2</td><td>16.5 ± 0.2</td><td>4.0±0.8</td><td>212</td><td>0.11</td><td>0.070</td></tr><tr><td>GraphCNN</td><td>MP</td><td>3.5± 0.4</td><td>15.0 ± 0.3</td><td>10±10</td><td>99</td><td>0.26</td><td>0.038</td></tr><tr><td>TreeGAN</td><td>MP</td><td>4.8±0.2</td><td>33±6</td><td>10±2</td><td>148</td><td>0.22</td><td>0.041</td></tr><tr><td>MP</td><td>FC</td><td>1.3± 0.1</td><td>4.5± 0.4</td><td>2.2 ±0.6</td><td>41</td><td>0.37</td><td>0.030</td></tr><tr><td>MP</td><td>PointNet</td><td>6.5± 0.3</td><td>23.2±0.6</td><td>6±1</td><td>850</td><td>0.18</td><td>0.034</td></tr><tr><td></td><td>FC</td><td></td><td></td><td></td><td></td><td>0.28</td><td>0.103</td></tr><tr><td rowspan=\"14\">Top quark</td><td>FC GraphCNN</td><td></td><td>4.8±0.3</td><td>14.5 ± 0.6</td><td>23±3</td><td>160</td><td></td><td>0.081</td></tr><tr><td>TreeGAN</td><td>FC</td><td>7.0±0.3</td><td>8.0±0.5</td><td>1k ±6k</td><td>15</td><td>0.48</td><td></td></tr><tr><td></td><td>FC</td><td>17.0 ± 0.2</td><td>19.6± 0.6</td><td>33±2</td><td>77</td><td>0.39</td><td>0.083</td></tr><tr><td>FC GraphCNN</td><td>PointNet PointNet</td><td>2.7± 0.1</td><td>1.6 ± 0.4</td><td>7.7 ±0.5</td><td>3.9</td><td>0.56</td><td>0.075 0.085</td></tr><tr><td>TreeGAN</td><td>PointNet</td><td>11.3 ± 0.9 5.19 ± 0.08</td><td>30±10 9.1 ± 0.3</td><td>37±2</td><td>30k 17</td><td>0.39 0.53</td><td>0.079</td></tr><tr><td>MP</td><td></td><td></td><td></td><td>16±2</td><td></td><td></td><td></td></tr><tr><td>MP-LFC</td><td>MP</td><td>0.6±0.2</td><td>2.3±0.3</td><td>2±1</td><td>0.37</td><td>0.57</td><td>0.071</td></tr><tr><td></td><td>MP</td><td>0.9±0.3</td><td>2.2±0.7</td><td>2±1</td><td>0.93</td><td>0.56</td><td>0.073</td></tr><tr><td>FC</td><td>MP</td><td>6.9 ± 0.1</td><td>39.1± 0.3</td><td>15±1</td><td>81</td><td>0.26</td><td>0.120</td></tr><tr><td>GraphCNN</td><td>MP</td><td>6.7±0.1</td><td>8.2±0.5</td><td>40±10</td><td>15</td><td>0.49</td><td>0.081</td></tr><tr><td>TreeGAN</td><td>MP</td><td>13.4 ± 0.4</td><td>45±7</td><td>50±30</td><td>66</td><td>0.29</td><td>0.101</td></tr><tr><td>MP</td><td>FC</td><td>12.9 ± 0.3</td><td>26.3± 0.4</td><td>46±3</td><td>58</td><td>0.27</td><td>0.103</td></tr><tr><td>MP</td><td>PointNet</td><td>0.76±0.08</td><td>1.6 ± 0.4</td><td>4±1</td><td>3.7</td><td>0.59</td><td>0.072</td></tr></table>",
+        "bbox": [
+            176,
+            119,
+            823,
+            757
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            814,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We additionally perform a latency measurement and find, using an NVIDIA A100 GPU, that MPGAN generation requires $3 5 . 7 \\mu \\mathrm { s }$ per jet. In comparison, the traditional generation process for JetNet is measured on an 8-CPU machine as requiring 46ms per jet, meaning MPGAN provides a three-ordersof-magnitude speed-up. Furthermore, as noted in App. B, the generation of JetNet is significantly simpler than full simulation and reconstruction used at the LHC, which has been measured to require 12.3s [44] and 4s [45] respectively per top quark jet. Hence in practical applications we anticipate MPGAN’s improvement to potentially rise to five-orders-of-magnitude. ",
+        "bbox": [
+            174,
+            92,
+            825,
+            188
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Real baseline comparison. We find that MPGAN’s jet-level $W _ { 1 }$ scores all fall within error of the baselines in Table 1, while those of alternative generators are several standard deviations away. This is particularly an issue with complex top quark particle clouds, where we can see in Fig. 3 none of the existing generators are able to learn the bimodal jet feature distributions, and smaller light quark clouds, where we see distortion of jet features due to difficulty reproducing the zero-padded particle features. No model is able to achieve particle-level scores close to the baseline, and only those of the $\\mathrm { F C } +$ PointNet combination and MPGAN are of the same order of magnitude. We conclude that MPGAN reproduces the physical observable distributions to the highest degree of accuracy, but note, however, that it requires further improvement in particle feature reconstruction before it is ready for practical application in HEP. ",
+        "bbox": [
+            174,
+            204,
+            825,
+            342
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Architecture discussion. To disentangle the effectiveness of the MP generator and discriminator, we train each individually with alternative counterparts (Table 2). With the same PointNet discriminator, the GraphCNN and TreeGAN generators perform worse than the simple FC generator for every metric on all three datasets. The physics-motivated MP generator on the other hand outperforms all on the gluon and top quark datasets, and significantly so on the jet-level $W _ { 1 }$ scores and the FPND. We note, however, that the MP generator is not a significant improvement over the other generators with an FC discriminator. Holding the generator fixed, the PointNet discriminator performs significantly better over the FC for all metrics. With the FC, GraphCNN, and TreeGAN generators, PointNet is also an improvement over the MP discriminator. With an MP generator, the MP discrimimator is more performant on jet-level $W _ { 1 }$ and FPND scores but, on the top quark dataset, degrades $W _ { 1 } ^ { \\mathrm { P } }$ relative to PointNet. ",
+        "bbox": [
+            174,
+            357,
+            825,
+            508
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We learn from these three things: (1) a generator or discriminator architecture is only as effective as its counterpart—even though the MPGAN combination is the best overall, when paired with a network which is not able to learn complex substructure, or which breaks the permutation symmetry, neither the generator or discriminator is performant, (2) for high-fidelity jet feature reconstruction, both networks must be able to learn complex multi-particle correlations—however, this can come at the cost of low-level feature accuracy, and (3) MPGAN’s masking strategy is highly effective as both MP networks are improvements all around on light quark jets. ",
+        "bbox": [
+            174,
+            516,
+            825,
+            613
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Particle cloud evaluation metrics. We now discuss the merits of each evaluation metrics and provide suggestions for their use in future work. Fig. 4 shows correlation plots between chosen pairs of our evaluation metrics. As expected, we find W1-M and W1-EFP to be highly correlated, as they both measure learning of global jet features. For rigorous validation we suggest measuring both but for time-sensitive use-cases, such as quick evaluations during model training, W1-M should be sufficient. W1-M, FPND, and W1-P are all measuring different aspects of the generation and are relatively uncorrelated. We expect FPND overall to be the best and most discriminatory metric for evaluation, as it compares features found by a SOTA classifier to be statistically optimum for characterizing jets, while the W1 scores are valuable for their interpretability. Out of these, W1-M/W1-EFP are the most important from a physics-standpoint, as we generally characterize collisions by the high-level features of the output jets, rather than the individual particle features. ",
+        "bbox": [
+            174,
+            628,
+            825,
+            780
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "MMD and coverage are both valuable for specifically evaluating the quality and diversity of samples respectively, however we see from Fig. 4 that they saturate after a certain point, after which FPND and $W _ { 1 }$ scores are necessary for stronger discrimination. We also note that in Table 2, models with low $W _ { 1 }$ scores relative to the baseline have the best coverage and MMD scores as well. This indicates that the $W _ { 1 }$ metrics are sensitive to both mode collapse (measured by coverage), which is expected as in terms of feature distributions mode collapse manifests as differing supports, to which the $W _ { 1 }$ distance is sensitive, as well as to individual sample quality (measured by MMD), which supports our claim that recovering jet feature distributions implies accurate learning of individual cloud structure. Together this suggests that low $W _ { 1 }$ scores are able validate sample quality and against mode collapse, and justifies our criteria that a practical ML simulation alternative have $W _ { 1 }$ scores close to the baselines in Table 2. In conclusion, for thorough validation of generated particle clouds, we recommend considering all three W-1 scores in conjunction with FPND, while MMD and coverage, being focused tests of these aspects of generation, may be useful for understanding failure modes during model development. ",
+        "bbox": [
+            174,
+            786,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/92f879a528070bea4a2d1e4d4e65954f0e87a6f8dd0b9cb84a91f341143224dc.jpg",
+        "image_caption": [
+            "Figure 4: Correlation plots between pairs of evaluation metrics, evaluated on 400 separate batches of 50,000 MPGAN generated top quark jets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            87,
+            826,
+            375
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            455,
+            823,
+            525
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Summary ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            561,
+            287,
+            579
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this work, we publish JetNet: a novel particle cloud dataset to advance machine learning (ML) research in high energy physics (HEP), and provide a novel point-cloud-style dataset containing rich underlying physics for the ML community to experiment with. We apply existing state-of-the-art point cloud generative models to JetNet, and propose several physics- and computer-vision-inspired metrics to rigorously evaluate generated clouds. We find that existing models are not performant on a number of metrics, and fail to reproduce high-level jet features—arguably the most significant aspect for HEP. Our new message-passing generative adversarial network (MPGAN) model, designed to capture complex global structure and handle variable-sized clouds significantly improves performance in this area, as well as other metrics. We propose MPGAN as a new baseline model on JetNet and invite others to improve upon it. ",
+        "bbox": [
+            173,
+            603,
+            825,
+            741
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Impact With our JetNet dataset and library, we hope to lower the barrier to entry, improve reproducibility, and encourage development in HEP and ML. Particularly so in the area of simulation, where an accurate and fast ML particle cloud generator will have significant impact in (1) lowering the computational and energy cost of HEP research, as well as (2) increasing precision and sensitivity to new physics at the Large Hadron Collider and future colliders by providing more high-quality simulated data samples. One negative consequence of this, however, may be a loss of interpretability, and hence trustability, of the particle production generative model, which may ultimately increase uncertainties—though the metrics we propose should mitigate against this. More broadly, further advancements in the field of ML point cloud generation may result in fake visual data generation for proliferation of misinformation and impersonation/identity theft. ",
+        "bbox": [
+            174,
+            772,
+            825,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            88,
+            553,
+            107
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 772369). R. K. was partially supported by an IRIS-HEP fellowship through the U.S. National Science Foundation (NSF) under Cooperative Agreement OAC-1836650, and by the LHC Physics Center at Fermi National Accelerator Laboratory, managed and operated by Fermi Research Alliance, LLC under Contract No. DE-AC02- 07CH11359 with the U.S. Department of Energy (DOE). J. D. is supported by the DOE, Office of Science, Office of High Energy Physics Early Career Research program under Award No. DESC0021187 and by the DOE, Office of Advanced Scientific Computing Research under Award No. DE-SC0021396 (FAIR4HEP). B. O and T. T are supported by grant 2018/25225-9, São Paulo Research Foundation (FAPESP). B. O was also partially supported by grants #2018/01398-1 and #2019/16401-0, São Paulo Research Foundation (FAPESP). J-R. V. is partially supported by the ERC under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 772369) and by the DOE, Office of Science, Office of High Energy Physics under Award No. DE-SC0011925, DE-SC0019227, and DE-AC02-07CH11359. D. G. is partially supported by the EU ICT-48 2020 project TAILOR (No. 952215). This work was performed using the Pacific Research Platform Nautilus HyperCluster supported by NSF awards CNS-1730158, ACI1540112, ACI-1541349, OAC-1826967, the University of California Office of the President, and the University of California San Diego’s California Institute for Telecommunications and Information Technology/Qualcomm Institute. Thanks to CENIC for the 100 Gpbs networks. Funding for cloud credits was supported by NSF Award #1904444 Internet2 supported E-CAS Exploring Clouds to Accelerate Science. ",
+        "bbox": [
+            173,
+            119,
+            826,
+            410
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            430,
+            266,
+            446
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[1] A. J. Larkoski, I. Moult, and B. Nachman, “Jet Substructure at the Large Hadron Collider: A Review of Recent Advances in Theory and Machine Learning”, Phys. Rept. 841 (2020) 1, doi:10.1016/j.physrep.2019.11.001, arXiv:1709.04464.   \n[2] L. de Oliveira, M. Paganini, and B. Nachman, “Learning particle physics by example: Location-aware generative adversarial networks for physics synthesis”, Comput. Softw. Big Sci. 1 (2017) 4, doi:10.1007/s41781-017-0004-6, arXiv:1701.05927.   \n[3] R. Kansal, “JETNET Library”. doi:10.5281/zenodo.5597893, https://github.com/jet-net/JetNet.   \n[4] R. K. Ellis, W. J. Stirling, and B. R. Webber, “QCD and collider physics”. Cambridge monographs on particle physics, nuclear physics, and cosmology. Cambridge University Press, Cambridge, 2003. Photography by S. Vascotto. doi:10.1017/CBO9780511628788.   \n[5] T. Sjöstrand et al., “An introduction to PYTHIA 8.2”, Comput. Phys. Commun. 191 (2015) 159, doi:10.1016/j.cpc.2015.01.024, arXiv:1410.3012.   \n[6] J. Bellm et al., “HERWIG 7.0/HERWIG $+ + 3 . 0$ release note”, Eur. Phys. J. C 76 (2016), no. 4, 196, doi:10.1140/epjc/s10052-016-4018-8, arXiv:1512.01178.   \n[7] B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, “Parton Fragmentation and String Dynamics”, Phys. Rept. 97 (1983) 31, doi:10.1016/0370-1573(83)90080-7.   \n[8] J. Cogan, M. Kagan, E. Strauss, and A. Schwarztman, “Jet-Images: Computer Vision Inspired Techniques for Jet Tagging”, JHEP 02 (2015) 118, doi:10.1007/JHEP02(2015)118, arXiv:1407.5675.   \n[9] L. de Oliveira et al., “Jet-images — deep learning edition”, JHEP 07 (2016) 069, doi:10.1007/JHEP07(2016)069, arXiv:1511.05190.   \n[10] H. Qu and L. Gouskos, “ParticleNet: Jet Tagging via Particle Clouds”, Phys. Rev. D 101 (2020), no. 5, 056019, doi:10.1103/PhysRevD.101.056019, arXiv:1902.08570.   \n[11] J. Shlomi, P. Battaglia, and J.-R. Vlimant, “Graph neural networks in particle physics”, Mach. Learn.: Sci. Technol. 2 (2021) 021001, doi:10.1088/2632-2153/abbf9a, arXiv:2007.13681.   \n[12] J. Duarte and J.-R. Vlimant, “Graph neural networks for particle tracking and reconstruction”, in Artificial Intelligence for High Energy Physics, P. Calafiura, D. Rousseau, and K. Terao, eds. World Scientific Publishing, 12, 2020. arXiv:2012.01249. Submitted to Int. J. Mod. Phys. A. doi:10.1142/12200.   \n[13] CMS Collaboration, “Identification of heavy, energetic, hadronically decaying particles using machine-learning techniques”, JINST 15 (2020) P06005, doi:10.1088/1748-0221/15/06/P06005, arXiv:2004.08262.   \n[14] A. X. Chang et al., “ShapeNet: An information-rich 3D model repository”, Technical Report, 2015. arXiv:1512.03012.   \n[15] R. Kansal et al., “JetNet”, 05, 2021. doi:10.5281/zenodo.4834876.   \n[16] M. Pierini, J. M. Duarte, N. Tran, and M. Freytsis, “hls4ml LHC jet dataset (30 particles)”, 01, 2020. doi:10.5281/zenodo.3601436.   \n[17] M. Paganini, L. de Oliveira, and B. Nachman, “CaloGAN $:$ Simulating 3D high energy particle showers in multilayer electromagnetic calorimeters with generative adversarial networks”, Phys. Rev. D 97 (2018), no. 1, 014021, doi:10.1103/PhysRevD.97.014021, arXiv:1712.10321.   \n[18] ATLAS Collaboration, “Fast simulation of the ATLAS calorimeter system with generative adversarial networks”, ATLAS Software Public Note ATL-SOFT-PUB-2020-006, 2020.   \n[19] M. Erdmann, J. Glombitza, and T. Quast, “Precise simulation of electromagnetic calorimeter showers using a Wasserstein generative adversarial network”, Comput. Softw. Big Sci. 3 (2019) 4, doi:10.1007/s41781-018-0019-7, arXiv:1807.01954.   \n[20] F. Carminati et al., “Generative adversarial networks for fast simulation”, J. Phys. Conf. Ser. 1525 (2020) 012064, doi:10.1088/1742-6596/1525/1/012064.   \n[21] Y. Lu, J. Collado, D. Whiteson, and P. Baldi, “Sparse autoregressive models for scalable generation of sparse images in particle physics”, Phys. Rev. D 103 (2021), no. 3, 036012, doi:10.1103/PhysRevD.103.036012, arXiv:2009.14017.   \n[22] S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift”, in Proceedings of the 32nd International Conference on Machine Learning, F. Bach and D. Blei, eds., volume 37, p. 448. PMLR, 2015. arXiv:1502.03167.   \n[23] C.-L. Liet al., “Point cloud GAN”, in 2019 ICLR Workshop Deep Generative Models for Highly Structured Data. 2018. arXiv:1810.05795.   \n[24] G. Yang et al., “PointFlow: 3D point cloud generation with continuous normalizing flows”, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), p. 4540. 2019. arXiv:1906.12320. doi:10.1109/ICCV.2019.00464.   \n[25] R. Klokov, E. Boyer, and J. Verbeek, “Discrete point flow networks for efficient point cloud generation”, in 2020 European Conference on Computer Vision (ECCV), A. Vedaldi, H. Bischof, T. Brox, and J.-M. Frahm, eds., p. 694. Springer International Publishing, Cham, 2020. arXiv:2007.10170.   \n[26] R. Cai et al., “Learning gradient fields for shape generation”, in 2020 European Conference on Computer Vision (ECCV), A. Vedaldi, H. Bischof, T. Brox, and J.-M. Frahm, eds., p. 364. Cham, 2020. arXiv:2008.06520.   \n[27] J. Köhler, L. Klein, and F. Noe, “Equivariant flows: Exact likelihood generative learning for symmetric densities”, in Proceedings of the 37th International Conference on Machine Learning, H. D. III and A. Singh, eds., volume 119 of Proceedings of Machine Learning Research, p. 5361. PMLR, 2020. arXiv:2006.02425.   \n[28] G. N. C. Simm, R. Pinsler, G. Csányi, and J. M. Hernández-Lobato, “Symmetry-aware actor-critic for 3d molecular design”, in International Conference on Learning Representations. 2021.   \n[29] N. Gebauer, M. Gastegger, and K. Schütt, “Symmetry-adapted generation of 3D point sets for the targeted discovery of molecules”, in Advances in Neural Information Processing Systems 32, H. Wallach et al., eds., p. 7566. Curran Associates, Inc., 2019. arXiv:1906.00957.   \n[30] P. Achlioptas, O. Diamanti, I. Mitliagkas, and L. Guibas, “Learning representations and generative models for 3D point clouds”, in Proceedings of the 35th International Conference on Machine Learning, J. Dy and A. Krause, eds., volume 80, p. 40. PMLR, 2018.   \n[31] D. Valsesia, G. Fracastoro, and E. Magli, “Learning localized generative models for 3D point clouds via graph convolution”, in International Conference on Learning Representations (ICLR) 2019. 2019.   \n[32] D. W. Shu, S. W. Park, and J. Kwon, “3D point cloud generative adversarial network based on tree structured graph convolutions”, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), p. 3858. 2019. arXiv:1905.06292. doi:10.1109/ICCV.2019.00396.   \n[33] C. R. Qi, H. Su, K. Mo, and L. J. Guibas, “PointNet: Deep learning on point sets for 3D classification and segmentation”, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). July, 2017.   \n[34] H. Wang et al., “Rethinking sampling in 3D point cloud generative adversarial networks”, 2020. arXiv:2006.07029.   \n[35] J. Gilmeret al., “Neural message passing for quantum chemistry”, in Proceedings of the 34th International Conference on Machine Learning, D. Precup and Y. W. Teh, eds., volume 70, p. 1263. PMLR, 2017. arXiv:1704.01212.   \n[36] CMS Collaboration, “Recent Developments in CMS Fast Simulation”, PoS ICHEP2016 (2016) 181, doi:10.22323/1.282.0181, arXiv:1701.03850.   \n[37] P. T. Komiske, E. M. Metodiev, and J. Thaler, “Energy flow polynomials: A complete linear basis for jet substructure”, JHEP 04 (2018) 013, doi:10.1007/JHEP04(2018)013, arXiv:1712.07124.   \n[38] A. Borji, “Pros and cons of GAN evaluation measures: New developments”, 2021. arXiv:2103.09396.   \n[39] M. Heuselet al., “GANs trained by a two time-scale update rule converge to a local nash equilibrium”, in Advances in Neural Information Processing Systems, I. Guyon et al., eds., volume 30, p. 6626. Curran Associates, Inc., 2017. arXiv:1706.08500.   \n[40] P. T. Komiske, E. M. Metodiev, and J. Thaler, “Metric Space of Collider Events”, Phys. Rev. Lett. 123 (2019), no. 4, 041801, doi:10.1103/PhysRevLett.123.041801, arXiv:1902.02346.   \n[41] Y. Wang et al., “Dynamic graph CNN for learning on point clouds”, ACM Trans. Graph. 38 (2019) doi:10.1145/3326362, arXiv:1801.07829.   \n[42] F. Monti et al., “Geometric deep learning on graphs and manifolds using mixture model CNNs”, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), p. 5425. 2017. arXiv:1611.08402. doi:10.1109/CVPR.2017.576.   \n[43] R. Kansal, “MPGAN Code”. doi:10.5281/zenodo.5598407, https://github.com/rkansal47/MPGAN/tree/neurips21.   \n[44] Pedro, Kevin, “Current and future performance of the cms simulation”, EPJ Web Conf. 214 (2019) 02036, doi:10.1051/epjconf/201921402036.   \n[45] C. Chen et al., “Data augmentation at the LHC through analysis-specific fast simulation with deep learning”, 2020. arXiv:2010.01835.   \n[46] K. Cranmer, J. Brehmer, and G. Louppe, “The frontier of simulation-based inference”, PNAS 117 (2020) 30055, doi:10.1073/pnas.1912789117, arXiv:1911.01429.   \n[47] G. Cowan, “Statistics”, Ch. 40 in Particle Data Group, P. A. Zyla et al., “Review of Particle Physics”, PTEP 2020 (2020) 083C01, doi:10.1093/ptep/ptaa104.   \n[48] CMS Collaboration, “Precise determination of the mass of the Higgs boson and tests of compatibility of its couplings with the standard model predictions using proton collisions at 7 and 8 TeV”, Eur. Phys. J. C 75 (2015) 212, doi:10.1140/epjc/s10052-015-3351-7, arXiv:1412.8662.   \n[49] R. J. Barlow and C. Beeston, “Fitting using finite Monte Carlo samples”, Comput. Phys. Commun. 77 (1993) 219, doi:10.1016/0010-4655(93)90005-W.   \n[50] J. Brehmer and K. Cranmer, “Simulation-based inference methods for particle physics”, in Artificial Intelligence for High Energy Physics, P. Calafiura, D. Rousseau, and K. Terao, eds. World Scientific Publishing, 10, 2020. arXiv:2010.06439. Submitted to Int. J. Mod. Phys. A. doi:10.1142/12200.   \n[51] J. Alwall et al., “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations”, JHEP 07 (2014) 079, doi:10.1007/JHEP07(2014)079, arXiv:1405.0301.   \n[52] R. D. Ball et al., “Parton distributions with LHC data”, Nucl. Phys. B 867 (2013) 244–289, doi:10.1016/j.nuclphysb.2012.10.003, arXiv:1207.1303.   \n[53] P. Skands, S. Carrazza, and J. Rojo, “Tuning PYTHIA 8.1: The Monash 2013 Tune”, Eur. Phys. J. C 74 (2014), no. 8, 3024, doi:10.1140/epjc/s10052-014-3024-y, arXiv:1404.5630.   \n[54] M. Cacciari, G. P. Salam, and G. Soyez, “The anti- $\\cdot k _ { \\mathrm { T } }$ jet clustering algorithm”, JHEP 04 (2008) 063, doi:10.1088/1126-6708/2008/04/063, arXiv:0802.1189.   \n[55] M. Cacciari, G. P. Salam, and G. Soyez, “FastJet user manual”, Eur. Phys. J. C 72 (2012) 1896, doi:10.1140/epjc/s10052-012-1896-2, arXiv:1111.6097.   \n[56] M. Cacciari and G. P. Salam, “Dispelling the $n ^ { 3 }$ myth for the $k _ { \\mathrm { T } }$ jet-finder”, Phys. Lett. B 641 (2006) 57, doi:10.1016/j.physletb.2006.08.037, arXiv:hep-ph/0512210.   \n[57] X. Mao et al., “Multi-class generative adversarial networks with the L2 loss function”, 2016. arXiv:1611.04076.   \n[58] N. Srivastavaet al., “Dropout: A simple way to prevent neural networks from overfitting”, J. Mach. Learn. Res. 15 (2014) 1929.   \n[59] I. Gulrajaniet al., “Improved training of Wasserstein GANs”, in Advances in Neural Information Processing Systems, I. Guyon et al., eds., volume 30, p. 5767. Curran Associates, Inc., 2017. arXiv:1704.00028.   \n[60] T. Miyato, T. Kataoka, M. Koyama, and Y. Yoshida, “Spectral normalization for generative adversarial networks”, in 6th International Conference on Learning Representations. 2018. arXiv:1802.05957.   \n[61] F. Schäfer, H. Zheng, and A. Anandkumar, “Implicit competitive regularization in GANs”, 2019. arXiv:1910.05852.   \n[62] T. Karraset al., “Training generative adversarial networks with limited data”, in Advances in Neural Information Processing Systems, H. Larochelle et al., eds., volume 33, p. 12104. Curran Associates, Inc., 2020. arXiv:2006.06676.   \n[63] N.-T. Tran et al., “On data augmentation for GAN training”, IEEE Trans. Image Process. 30 (2021) 1882, doi:10.1109/tip.2021.3049346, arXiv:2006.05338.   \n[64] Z. Zhao et al., “Image augmentations for GAN training”, 2020. arXiv:2006.02595.   \n[65] M. Arjovsky, S. Chintala, and L. Bottou, “Wasserstein GAN”, 2017. arXiv:1701.07875. ",
+        "bbox": [
+            179,
+            453,
+            828,
+            911
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            56,
+            828,
+            916
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            68,
+            828,
+            917
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            89,
+            828,
+            627
+        ],
+        "page_idx": 13
+    }
+]

parse/train/r1gIa0NtDH/r1gIa0NtDH.md

@@ -0,0 +1,327 @@
+# MELNET: A GENERATIVE MODEL FOR AUDIO IN THE FREQUENCY DOMAIN
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Capturing high-level structure in audio waveforms is challenging because a single second of audio spans tens of thousands of timesteps. While long-range dependencies are difficult to model directly in the time domain, we show that they can be more tractably modelled in two-dimensional time-frequency representations such as spectrograms. By leveraging this representational advantage, in conjunction with a highly expressive probabilistic model and a multiscale generation procedure, we design a model capable of generating high-fidelity audio samples which capture structure at timescales which time-domain models have yet to achieve. We demonstrate that our model captures longer-range dependencies than time-domain models such as WaveNet across a diverse set of unconditional generation tasks, including single-speaker speech generation, multi-speaker speech generation, and music generation.
+
+# 1 INTRODUCTION
+
+Audio waveforms have complex structure at drastically varying timescales, which presents a challenge for generative models. Local structure must be captured to produce high-fidelity audio, while longrange dependencies spanning tens of thousands of timesteps must be captured to generate audio which is globally consistent. Existing generative models of waveforms such as WaveNet (van den Oord et al., 2016a) and SampleRNN (Mehri et al., 2016) are well-adapted to model local dependencies, but as these models typically only backpropagate through a fraction of a second, they are unable to capture high-level structure that emerges on the scale of several seconds.
+
+We introduce a generative model for audio which captures longer-range dependencies than existing end-to-end models. We primarily achieve this by modelling 2D time-frequency representations such as spectrograms rather than 1D time-domain waveforms (Figure 1). The temporal axis of a spectrogram is orders of magnitude more compact than that of a waveform, meaning dependencies that span tens of thousands of timesteps in waveforms only span hundreds of timesteps in spectrograms. In practice, this enables our spectrogram models to generate unconditional speech and music samples with consistency over multiple seconds whereas time-domain models must be conditioned on intermediate features to capture structure at similar timescales.
+
+Modelling spectrograms can simplify the task of capturing global structure, but can weaken a model’s ability to capture local characteristics that correlate with audio fidelity. Producing high-fidelity audio has been challenging for existing spectrogram models, which we attribute to the lossy nature of spectrograms and oversmoothing artifacts which result from insufficiently expressive models. To reduce information loss, we model high-resolution spectrograms which have the same dimensionality as their corresponding time-domain signals. To limit oversmoothing, we use a highly expressive autoregressive model which factorizes the distribution over both the time and frequency dimensions.
+
+Modelling both fine-grained details and high-level structure in high-dimensional distributions is known to be challenging for autoregressive models. To capture both local and global structure in spectrograms with hundreds of thousands of dimensions, we employ a multiscale approach which generates spectrograms in a coarse-to-fine manner. A low-resolution, subsampled spectrogram that captures high-level structure is generated initially, followed by an iterative upsampling procedure that adds high-resolution details.
+
+![](images/cc6743f8d442787f246b62a0186760052782c67babcb05064f0a46a3806f16d8.jpg)  
+Figure 1: Spectrogram and waveform representations of the same 4 second audio signal. The waveform spans nearly 100,000 timesteps whereas the temporal axis of the spectrogram spans roughly 400. Complex structure is nested within the temporal axis of the waveform at various timescales, whereas the spectrogram has structure which is smoothly spread across the time-frequency plane.
+
+Combining these representational and modelling techniques yields a highly expressive and broadly applicable generative model of audio. Our contributions are are as follows:
+
+• We introduce MelNet, a generative model for spectrograms which couples a fine-grained autoregressive model and a multiscale generation procedure to jointly capture local and global structure. We show that MelNet is able to model longer-range dependencies than existing time-domain models. Additionally, we include an ablation to demonstrate that multiscale modelling is essential for modelling long-range dependencies.   
+We demonstrate that MelNet is broadly applicable to a variety of audio generation tasks, including unconditional speech and music generation. Furthermore, MelNet is able to model highly multimodal data such as multi-speaker and multilingual speech.
+
+# 2 PRELIMINARIES
+
+We briefly present background regarding spectral representations of audio. Audio is represented digitally as a one-dimensional, discrete-time signal $y = ( y _ { 1 } , \dots , y _ { n } )$ . Existing generative models for audio have predominantly focused on modelling these time-domain signals directly. We instead model spectrograms, which are two-dimensional time-frequency representations which contain information about how the frequency content of an audio signal varies through time. Spectrograms are computed by taking the squared magnitude of the short-time Fourier transform (STFT) of a time-domain signal, i.e. $x = \| \mathrm { S T F T } ( y ) \| ^ { 2 }$ . The value of $x _ { i j }$ (referred to as amplitude or energy) corresponds to the squared magnitude of the $j$ th element of the frequency response at timestep i. Each slice $x _ { i , * }$ is referred to as a frame. We assume a time-major ordering, but following convention, all figures are displayed transposed and with the frequency axis inverted.
+
+Time-frequency representations such as spectrograms highlight how the tones and pitches within an audio signal vary through time. Such representations are closely aligned with how humans perceive audio. To further align these representations with human perception, we convert the frequency axis to the Mel scale and apply an elementwise logarithmic rescaling of the amplitudes. Roughly speaking, the Mel transformation aligns the frequency axis with human perception of pitch and the logarithmic rescaling aligns the amplitude axis with human perception of loudness.
+
+Spectrograms are lossy representations of their corresponding time-domain signals. The Mel transformation discards frequency information and the removal of the STFT phase discards temporal information. When recovering a time-domain signal from a spectrogram, this information loss manifests as distortion in the recovered signal. To minimize these artifacts and improve the fidelity of generated audio, we model high-resolution spectrograms. The temporal resolution of a spectrogram can be increased by decreasing the STFT hop size, and the frequency resolution can be increased by increasing the number of Mel channels. Generated spectrograms are converted back to time-domain signals using classical spectrogram inversion algorithms. We experiment with both Griffin-Lim (Griffin & Lim, 1984) and a gradient-based inversion algorithm (Decorsiere et al., 2015), and ultimately \` use the latter as it generally produced audio with fewer artifacts.
+
+![](images/aa045a4ec7d0df28be1be4bc52a74c1d3b9eb7ed701c13f29cacc84fbac62140.jpg)  
+Figure 2: The context $x _ { < i j }$ (grey) for the element $\boldsymbol { x } _ { i j }$ (black) is encoded using 4 RNNs. Three of these are used in the time-delayed stack to extract features from preceding frames. The fourth is used in the frequency-delayed stack to extract features from all preceding elements within the current frame. Each arrow denotes an individual RNN cell and arrows of the same color use shared parameters.
+
+# 3 PROBABILISTIC MODEL
+
+We use an autoregressive model which factorizes the joint distribution over a spectrogram $x$ as a product of conditional distributions. Given an ordering of the dimensions of $x$ , we define the context $x _ { < i j }$ as the elements of $x$ that precede $x _ { i j }$ . We default to a row-major ordering which proceeds through each frame $x _ { i , * }$ from low to high frequency, before progressing to the next frame. The joint density is factorized as
+
+$$
+p ( \boldsymbol x ) = \prod _ { i } \prod _ { j } p ( x _ { i j } \mid \boldsymbol x _ { < i j } ; \theta _ { i j } ) ,
+$$
+
+where $\theta _ { i j }$ parameterizes a univariate density over $x _ { i j }$ . We model each factor distribution as a Gaussian mixture model with $K$ components. Thus, $\theta _ { i j }$ consists of $3 K$ parameters corresponding to means $\{ \mu _ { i j k } \} _ { k = 1 } ^ { K }$ , standard deviations $\{ \sigma _ { i j k } \} _ { k = 1 } ^ { K }$ , and mixture coefficients $\{ \pi _ { i j k } \} _ { k = 1 } ^ { K }$ . The resulting factor distribution can then be expressed as
+
+$$
+p ( x _ { i j } \mid x _ { < i j } ; \theta _ { i j } ) = \sum _ { k = 1 } ^ { K } \pi _ { i j k } \mathcal { N } ( x _ { i j } ; \mu _ { i j k } , \sigma _ { i j k } ) .
+$$
+
+Following the work on Mixture Density Networks (Bishop, 1994) and their application to autoregressive models (Graves, 2013), $\theta _ { i j }$ is modelled as the output of a neural network and computed as a function of the context $x _ { < i j }$ . Precisely, for some network $f$ with parameters $\psi$ , we have $\theta _ { i j } = f ( \boldsymbol x _ { < i j } ; \ \psi )$ . A maximum-likelihood estimate for the network parameters is computed by minimizing the negative log-likelihood via gradient descent.
+
+To ensure that the network output parameterizes a valid Gaussian mixture model, the network first computes unconstrained parameters $\{ \hat { \mu } _ { i j k } , \hat { \sigma } _ { i j k } , \hat { \pi } _ { i j k } \} _ { k = 1 } ^ { K }$ as a vector $\hat { \theta } _ { i j } ~ \in ~ \mathbb { R } ^ { 3 K }$ , and enforces constraints on $\theta _ { i j }$ by applying the following transformations:
+
+$$
+\begin{array} { r l } & { \mu _ { i j k } = \hat { \mu } _ { i j k } } \\ & { \sigma _ { i j k } = \exp ( \hat { \sigma } _ { i j k } ) } \\ & { \pi _ { i j k } = \cfrac { \exp \left( \hat { \pi } _ { i j k } \right) } { \sum _ { k = 1 } ^ { K } \exp \left( \hat { \pi } _ { i j k } \right) } . } \end{array}
+$$
+
+These transformations ensure the standard deviations $\sigma _ { i j k }$ are positive and the mixture coefficients $\pi _ { i j k }$ sum to one.
+
+# 4 NETWORK ARCHITECTURE
+
+To model the distribution in an autoregressive manner, we design a network which computes the distribution over $x _ { i j }$ as a function of the context $x _ { < i j }$ . The network architecture draws inspiration from existing autoregressive models for images (Theis & Bethge, 2015; van den Oord et al., 2016c;b; Chen et al., 2017; Salimans et al., 2017; Parmar et al., 2018; Child et al., 2019). In the same way that these models estimate a distribution pixel-by-pixel over the spatial dimensions of an image, our model estimates a distribution element-by-element over the time and frequency dimensions of a spectrogram. A noteworthy distinction is that spectrograms are not invariant to translation along the frequency axis, making 2D convolution less desirable than other 2D network primitives which do not assume invariance. Utilizing multidimensional recurrence instead of 2D convolution has been shown to be beneficial when modelling spectrograms in discriminative settings (Li et al., 2016; Sainath & Li, 2016), which motivates our use of an entirely recurrent architecture.
+
+Similar to Gated PixelCNN (van den Oord et al., 2016b), the network has multiple stacks of computation. These stacks extract features from different segments of the input to collectively summarize the full context x<ij :
+
+• The time-delayed stack computes features which aggregate information from all previous frames x<i,∗. • The frequency-delayed stack utilizes all preceding elements within a frame, $x _ { i , < j }$ , as well as the outputs of the time-delayed stack, to summarize the full context $x _ { < i j }$ .
+
+The stacks are connected at each layer of the network, meaning that the features generated by layer $l$ of the time-delayed stack are used as input to layer $l$ of the frequency-delayed stack. To facilitate the training of deeper networks, both stacks use residual connections (He et al., 2016). The outputs of the final layer of the frequency-delayed stack are used to compute the unconstrained parameters $\hat { \theta }$ .
+
+# 4.1 TIME-DELAYED STACK
+
+The time-delayed stack utilizes multiple layers of multidimensional RNNs to extract features from $x _ { < i , * }$ , the two-dimensional region consisting of all frames preceding $x _ { i j }$ . Each multidimensional RNN is composed of three one-dimensional RNNs: one which runs forwards along the frequency axis, one which runs backwards along the frequency axis, and one which runs forwards along the time axis. Each RNN runs along each slice of a given axis, as shown in Figure 2. The output of each layer of the time-delayed stack is the concatenation of the three RNN hidden states.
+
+We denote the function computed at layer $l$ of the time-delayed stack (three RNNs followed by concatenation) as $\mathcal { F } _ { l } ^ { t }$ . At each layer, the time-delayed stack uses the feature map from the previous layer, $h ^ { t } [ l - 1 ]$ , to compute the subsequent feature map $\mathcal { F } _ { l } ^ { t } \big ( h ^ { t } [ l - 1 ] \big )$ which consists of the three concatenated RNN hidden states. When using residual connections, the computation of $h ^ { t } [ l ]$ from $h ^ { t } [ l - 1 ]$ becomes
+
+$$
+h _ { i j } ^ { t } [ l ] = W _ { l } ^ { t } \mathcal { F } _ { l } ^ { t } \big ( h ^ { t } [ l - 1 ] \big ) _ { i j } + h _ { i j } ^ { t } [ l - 1 ] .
+$$
+
+To ensure the output $h _ { i j } ^ { t } [ l ]$ is only a function of frames which lie in the context $x _ { < i j }$ , the inputs to the time-delayed stack are shifted backwards one step in time: $h _ { i j } ^ { t } [ 0 ] = W _ { 0 } ^ { t } x _ { i - 1 , j }$ .
+
+# 4.2 FREQUENCY-DELAYED STACK
+
+The frequency-delayed stack is a one-dimensional RNN which runs forward along the frequency axis. Much like existing one-dimensional autoregressive models (language models, waveform models, etc.), the frequency-delayed stack operates on a one-dimensional sequence (a single frame) and estimates the distribution for each element conditioned on all preceding elements. The primary difference is that it is also conditioned upon the outputs of the time-delayed stack, allowing it to use the full two-dimensional context $x _ { < i j }$ .
+
+We denote the function computed by the frequency-delayed stack as $\mathcal { F } _ { l } ^ { f }$ . At each layer, the frequencydelayed stack takes two inputs: the the previous-layer outputs of the frequency-delayed stack, $h _ { i j } ^ { f } [ l - 1 ]$ , and the current-layer outputs of the time-delayed stack $h _ { i j } ^ { t } [ l ]$ . These inputs are summed and used as input to a one-dimensional RNN to produce the output feature map $\mathcal { F } _ { l } ^ { f } \left( h ^ { f } [ l - 1 ] , ~ h ^ { t } [ l ] \right)$ which consists of the RNN hidden state:
+
+$$
+h _ { i j } ^ { f } [ l ] = W _ { l } ^ { f } \mathcal { F } _ { l } ^ { f } \big ( h ^ { f } [ l - 1 ] , ~ h ^ { t } [ l ] \big ) _ { i j } + h _ { i j } ^ { f } [ l - 1 ] .
+$$
+
+![](images/0d7e0c8d009dd29dc0c0c3cb3679f00ab978dd75546732c36aaa4963fe08fbe5.jpg)  
+Figure 3: Computation graph for a single layer of the network. $\mathcal { F } _ { l } ^ { t }$ and $\mathcal { F } _ { l } ^ { f }$ are the functions computed by the time-delayed stack and frequency-delayed stack, respectively. The outputs of these functions are projected (by the matrices $\it { W } _ { l } ^ { t }$ and $\boldsymbol { W } _ { l } ^ { f }$ ) and summed with the layer inputs to form residual blocks.
+
+To ensure that $h _ { i j } ^ { f } [ l ]$ is computed using only elements in the context $x _ { < i j }$ , the inputs to the frequencydelayed stack are shifted backwards one step along the frequency axis: $h _ { i j } ^ { f } [ 0 ] = W _ { 0 } ^ { f } x _ { i , j - 1 }$ . At the final layer, layer $L$ , a linear map is applied to the output of the frequency-delayed stack to produce the unconstrained Gaussian mixture model parameters, i.e. $\hat { \theta } _ { i j } = \bar { W } _ { \theta } h _ { i j } ^ { f } [ L ]$ .
+
+# 4.3 CONDITIONING
+
+To incorporate conditioning information into the model, conditioning features $z$ are simply projected onto the input layer along with the inputs $x$ :
+
+$$
+\begin{array} { r l } & { h _ { i j } ^ { t } [ 0 ] = W _ { 0 } ^ { t } x _ { i - 1 , j } + W _ { z } ^ { t } z _ { i j } } \\ & { h _ { i j } ^ { f } [ 0 ] = W _ { 0 } ^ { f } x _ { i , j - 1 } + W _ { z } ^ { f } z _ { i j } . } \end{array}
+$$
+
+Reshaping, upsampling, and broadcasting can be used as necessary to ensure the conditioning features have the same time and frequency shape as the input spectrogram, e.g. a one-hot vector representation for speaker ID would first be broadcast along both the time and frequency axes.
+
+# 5 MULTISCALE MODELLING
+
+To improve audio fidelity, we generate high-resolution spectrograms which have the same dimensionality as their corresponding time-domain representations. Under this regime, a single training example has several hundreds of thousands of dimensions. Capturing global structure in such high-dimensional distributions is challenging for autoregressive models, which are biased towards capturing local dependencies. To counteract this, we utilize a multiscale approach which effectively permutes the autoregressive ordering so that a spectrogram is generated in a coarse-to-fine order.
+
+The elements of a spectrogram $x$ are partitioned into $G$ tiers $x ^ { 1 } , \ldots , x ^ { G }$ , such that each successive tier contains higher-resolution information. We define $x ^ { < g }$ as the union of all tiers which precede $x ^ { g }$ , i.e. $x ^ { < g } = ( x ^ { \top } , \ldots , x ^ { g - 1 } )$ . The distribution is factorized over tiers:
+
+$$
+p ( x ; \psi ) = \prod _ { g } p ( x ^ { g } \mid x ^ { < g } ; \psi ^ { g } ) ,
+$$
+
+and the distribution of each tier is further factorized element-by-element as described in Section 3. We explicitly include the parameterization by $\boldsymbol { \psi } = ( \psi ^ { 1 } , \dots , \psi ^ { G } )$ to indicate that each tier is modelled by a separate network.
+
+![](images/e5ad1d7dc090959b41614a197b15f6dc0e0ed81fefbd8a20c77ac7d878e9b434.jpg)  
+Figure 4: A sampled spectrogram viewed at different stages of the multiscale generation procedure. The initial tier dictates high-level structure and subsequent tiers add fine-grained details. Each upsampling tier doubles the resolution of the spectrogram.
+
+![](images/662744988b39d6711b2b2d20475d678c6e982221558b25fe7fa43b3f9f632c7c.jpg)  
+Figure 5: Schematic showing how tiers of the multiscale model are interleaved and used to condition the distribution for the subsequent tier. a) The initial tier is generated unconditionally. b) The second tier is generated conditionally given the the initial tier. c) The outputs of tiers 1 and 2 are interleaved along the frequency axis and used to condition the generation of tier 3. d) Tier 3 is interleaved along the time axis with all preceding tiers and used to condition the generation of tier 4.
+
+# 5.1 TRAINING
+
+During training, the tiers are generated by recursively partitioning a spectrogram into alternating rows along either the time or frequency axis. We define a function split which partitions an input into even and odd rows along a given axis. The initial step of the recursion applies the split function to a spectrogram $x$ , or equivalently $x ^ { < G + 1 }$ , so that the even-numbered rows are assigned to $x ^ { G }$ and the odd-numbered rows are assigned to $x ^ { < G }$ . Subsequent tiers are defined similarly in a recursive manner:
+
+$$
+x ^ { g } , x ^ { < g } = { \tt s p l i t } ( x ^ { < g + 1 } ) .
+$$
+
+At each step of the recursion, we model the distribution $p ( x ^ { g } \mid x ^ { < g } ; \psi ^ { g } )$ . The final step of the recursion models the unconditional distribution over the initial tier $p ( x ^ { 1 } ; \psi ^ { 1 } )$ .
+
+To model the conditional distribution $p ( x ^ { g } \mid x ^ { < g } ; \psi ^ { g } )$ , the network at each tier needs a mechanism to incorporate information from the preceding tiers $\scriptstyle { \dot { x } } ^ { < g }$ . To this end, we add a feature extraction network which computes features from $x ^ { < g }$ which are used condition the generation of $x ^ { g }$ . We use a multidimensional RNN consisting of four one-dimensional RNNs which run bidirectionally along slices of both axes of the context $x ^ { < g }$ . A layer of the feature extraction network is similar to a layer of the time-delayed stack, but since the feature extraction network is not causal, we include an RNN which runs backwards along the time axis and do not shift the inputs. The hidden states of the RNNs in the feature extraction network are used to condition the generation of $x ^ { g }$ . As each tier doubles the resolution, the features extracted from $x ^ { < g }$ have the same time and frequency shape as $x ^ { g }$ , allowing the conditioning mechanism described in section 4.3 to be used straightforwardly.
+
+# 5.2 SAMPLING
+
+To sample from the multiscale model we iteratively sample a value for $x ^ { g }$ conditioned on $x ^ { < g }$ using the learned distributions defined by the estimated network parameters $\hat { \psi } = ( \hat { \psi } ^ { 1 } , \dots , \hat { \psi } ^ { G } )$ . The initial tier, $x ^ { 1 }$ , is generated unconditionally by sampling from $p ( x ^ { 1 } ; \hat { \psi } ^ { 1 } )$ and subsequent tiers are sampled from $p ( x ^ { g } \mid x ^ { < g } ; { \hat { \psi } } ^ { g } )$ . At each tier, the sampled $x ^ { g }$ is interleaved with the context $x ^ { < g }$ :
+
+$$
+\begin{array} { r } { x ^ { < g + 1 } = \mathrm { i n t e r l e a v e } ( x ^ { g } , x ^ { < g } ) . } \end{array}
+$$
+
+The interleave function is simply the inverse of the split function. Sampling terminates once a full spectrogram, $x ^ { < G + 1 }$ , has been generated. A spectrogram generated by a multiscale model is shown in Figure 4 and the sampling procedure is visualized schematically in Figure 5.
+
+# 6 EXPERIMENTS
+
+To demonstrate the MelNet is broadly applicable as a generative model for audio, we train the model on a diverse set of audio generation tasks (single-speaker speech generation, multi-speaker speech generation, and music generation) using three publicly available datasets. Generated audio samples for each task are available on the accompanying web page https://audio-samples.github.io. We include samples generated using the priming and biasing procedures described by Graves (2013). Biasing lowers the temperature of the predictive distribution and priming seeds the model state with a given sequence of audio prior to sampling. Hyperparameters for all experiments are available in Appendix A.
+
+Speech and music have rich hierarchies of latent structure. Speech has complex linguistic structure (phonemes, words, syntax, semantics, etc.) and music has highly compositional musical structure (notes, chords, melody and rhythm, etc.). The presence of these latent structures in generated samples can be used as a proxy for how well a generative model has learned dependencies at various timescales. As such, a qualitative analysis of unconditional samples is an insightful method of evaluating generative models of audio. To facilitate such a qualitative evaluation, we train MelNet on each of the three unconditional generation tasks and include samples on the accompanying web page. For completeness, we briefly provide some of our own qualitative observations regarding the generated samples (Sections 6.1, 6.2, and 6.3). In addition to qualitative analysis, we conduct a human evaluation experiment to quantitatively compare how well WaveNet and MelNet capture high-level structure (Section 6.4). Lastly, we ablate the impact of the multiscale generation procedure on MelNet’s ability model long-range dependencies (Section 6.5).
+
+# 6.1 SINGLE-SPEAKER SPEECH
+
+To test MelNet’s ability to model a single speaker in a controlled environment, we utilize the Blizzard 2013 dataset (King, 2011), which consists of audiobook narration performed in a highly animated manner by a professional speaker. We find that MelNet frequently generates samples that contain coherent words and phrases. Even when the model generates incoherent speech, the intonation, prosody, and speaking style remain consistent throughout the duration of the sample. Furthermore, the model learns to produce speech using a variety of character voices and learns to generate samples which contain elements of narration and dialogue. Biased samples tend to contain longer strings of comprehensible words but are read in a less expressive fashion. When primed with a real sequence of audio, MelNet is able to continue sampling speech which has consistent speaking style and intonation.
+
+# 6.2 MULTI-SPEAKER SPEECH
+
+Audiobook data is recorded in a highly controlled environment. To demonstrate MelNet’s capacity to model distributions with significantly more variation, we utilize the VoxCeleb2 dataset (Chung et al., 2018). The VoxCeleb2 dataset consists of over 2,000 hours of speech data captured with real world noise including laughter, cross-talk, channel effects, music and other sounds. The dataset is also multilingual, with speech from speakers of 145 different nationalities, covering a wide range of accents, ages, ethnicities and languages. When trained on the VoxCeleb2 dataset, we find that MelNet is able to generate unconditional samples with significant variation in both speaker characteristics (accent, language, prosody, speaking style) as well as acoustic conditions (background noise and recording quality). While the generated speech is often not comprehensible, samples can often be identified as belonging to a specific language, indicating that the model has learned distinct modalities for different languages. Furthermore, it is difficult to distinguish real and fake samples which are spoken in foreign languages. For foreign languages, semantic structures are not understood by the listener and cannot be used to discriminate between real and fake. Consequently, the listener must rely largely on phonetic structure, which MelNet is able to realistically model.
+
+# 6.3 MUSIC
+
+To show that MelNet can model audio modalities other than speech, we apply the model to the task of unconditional music generation. We utilize the MAESTRO dataset (Hawthorne et al., 2018), which consists of over 172 hours of solo piano performances. The samples demonstrate that MelNet learns musical structures such as melody and harmony. Furthermore, generated samples often maintain consistent tempo and contain interesting variation in volume, timbre, and rhythm.
+
+<table><tr><td colspan="2">WaveNet</td><td>MelNet</td></tr><tr><td>Blizzard</td><td>0.0%</td><td>100.0%</td></tr><tr><td>VoxCeleb2</td><td>0.0%</td><td>100.0%</td></tr><tr><td>MAESTRO</td><td>4.2%</td><td>95.8%</td></tr></table>
+
+(a) Comparison between MelNet and WaveNet. Both models are trained in an entirely unsupervised manner.
+
+<table><tr><td colspan="2">Wave2Midi2Wave</td><td>MelNet</td></tr><tr><td>MAESTRO</td><td>37.7%</td><td>62.3 %</td></tr></table>
+
+(b) Comparison between MelNet and Wave2Midi2Wave. Wave2Midi2Wave is a two-stage model consisting of a Music Transformer trained on labelled MIDI followed by a conditional WaveNet model. The MelNet model, on the other hand, is trained without any intermediate supervision.
+
+Table 1: Selection rates of human evaluators when asked to identify which model generates samples with longer-term structure. Results show that MelNet captures long-range structure better than WaveNet. Furthermore, MelNet outperforms a two-stage model which conditions WaveNet on generated MIDI.
+
+# 6.4 HUMAN EVALUATION
+
+Making quantitative comparisons with existing generative models such as WaveNet is difficult for various reasons and previous works have ultimately relied on largely empirical evaluations by the reader (Dieleman et al., 2018). To allow the reader to make these judgements for themselves, we provide samples from both WaveNet and MelNet for each of the tasks described in the previous sections. Furthermore, in an effort to provide quantitative metrics to support the claim that MelNet generates samples with improved long-range structure in comparison to WaveNet, we conduct a human experiment whereby participants are presented anonymized samples from both models and asked to select which sample exhibits longer-term structure. We resort to such evaluations since standard metrics for evaluation of generative models such as density estimates cannot be used to compare WaveNet and MelNet as that these models operate on different representations.
+
+The methodology for this experiment is as follows. For each of the three unconditional audio generation tasks, we generated 50 samples from WaveNet and 50 samples from MelNet. Participants were shown an anonymized, randomly-drawn sample from each model and instructed to “select the sample which has more coherent long-term structure.” We collected 50 evaluations for each task. Results, shown in Table 1a, show that evaluators overwhelmingly agreed that samples generated by MelNet had more coherent long-range structure than samples from WaveNet across all tasks.
+
+In addition to comparing MelNet to an unconditional WaveNet model for music generation, we also compare to a two-stage Wave2Midi2Wave model (Hawthorne et al., 2018) which conditions WaveNet on MIDI generated by a separately-trained Music Transformer (Huang et al., 2018). The two-stage Wave2Midi2Wave model has the advantage of directly modelling labelled musical notes which distill much of the salient, high-level structure in music into a compact symbolic representation. Despite this, as shown by the results in Table 1b, the two-stage model does not capture long-range structure as well as a MelNet model that is trained without access to any intermediate representations.
+
+# 6.5 ABLATION: MULTISCALE MODELLING
+
+To isolate the impact of multiscale modelling procedure described in Section 5, we train models with varying numbers of tiers and evaluate the long-term coherence of their respective samples. As noted before, long-term coherence is difficult to quantify and we provide samples on the accompanying web page so that the reader can make their own judgements. We believe the samples clearly demonstrate that increasing the number of tiers results in samples with more coherent high-level structure. We note that our experiment varies the number of tiers from two to five. Training a single-tier model on full-resolution spectrograms was prohibitively expensive in terms of memory consumption. This highlights another benefit of multiscale modelling—large, deep networks can be allocated to learning complex distributional structure in the initial tiers while shallower networks can be used for modelling the relatively simple, low-entropy distributions in the upsampling tiers. This allows multiscale models to effectively allocate network capacity in proportion to the complexity of the modelling task.
+
+# 7 RELATED WORK
+
+The predominant line of research regarding generative models for audio has been directed towards modelling time-domain waveforms with autoregressive models (van den Oord et al., 2016a; Mehri et al., 2016; Kalchbrenner et al., 2018). WaveNet is a competitive baseline for audio generation, and as such, is used for comparison in many of our experiments. However, we note that the contribution of our work is in many ways complementary to that of WaveNet. MelNet is more proficient at capturing high-level structure, whereas WaveNet is capable of producing higher-fidelity audio. Several works have demonstrated that time-domain models can be used to invert spectral representations to highfidelity audio (Shen et al., 2018; Prenger et al., 2019; Arık et al., 2019), suggesting that MelNet could be used in concert with time-domain models such as WaveNet.
+
+Dieleman et al. (2018) and van den Oord et al. (2017) capture long-range dependencies in waveforms by utilizing a hierarchy of autoencoders. This approach requires multiple stages of models which must be trained sequentially, whereas the multiscale approach in this work can be parallelized over tiers. Additionally, these approaches do not directly optimize the data likelihood, nor do they admit tractable marginalization over the latent codes. We also note that the modelling techniques devised in these works can be broadly applied to autoregressive models such as ours, making their contributions largely complementary to ours.
+
+Recent works have used generative adversarial networks (GANs) (Goodfellow et al., 2014) to model both waveforms and spectral representations (Donahue et al., 2018; Engel et al., 2018). As with image generation, it remains unclear whether GANs capture all modes of the data distribution. Furthermore, these approaches are restricted to generating fixed-duration segments of audio, which precludes their usage in many audio generation tasks.
+
+Generating spectral representations is common practice for end-to-end text-to-speech models (Ping et al., 2017; Sotelo et al., 2017; Wang et al., 2017; Taigman et al., 2018). However, these models use probabilistic models which are much less expressive than the fine-grained autoregressive model used by MelNet. Consequently, these models are unsuitable for modelling high-entropy, multimodal distributions such as those involved in tasks like unconditional music generation.
+
+The network architecture used for MelNet is heavily influenced by recent advancements in deep autoregressive models for images. Theis & Bethge (2015) introduced an LSTM architecture for autoregressive modelling of 2D images and van den Oord et al. (2016c) introduced PixelRNN and PixelCNN and scaled up the models to handle the modelling of natural images. Subsequent works in autoregressive image modelling have steadily improved state-of-the-art for image density estimation (van den Oord et al., 2016b; Salimans et al., 2017; Parmar et al., 2018; Chen et al., 2017; Child et al., 2019). We draw inspiration from many of these models, and ultimately design a recurrent architecture of our own which is suitable for modelling spectrograms rather than images. We note that our choice of architecture is not a fundamental contribution of this work. While we have designed the architecture particularly for modelling spectrograms, we did not experimentally validate whether it outperforms existing architectures and make no such claims to this effect.
+
+We use a multidimensional recurrence in both the time-delayed stack and the upsampling tiers to extract features from two-dimensional inputs. Our multidimensional recurrence is effectively ‘factorized’ as it independently applies one-dimensional RNNs across each dimension. This approach differs from the tightly coupled multidimensional recurrences used by MDRNNs (Graves et al., 2007; Graves & Schmidhuber, 2009) and GridLSTMs (Kalchbrenner et al., 2015) and more closely resembles the approach taken by ReNet (Visin et al., 2015). Our approach allows for efficient training as we can extract features from an $M \times N$ grid in $\operatorname* { m a x } ( M , N )$ sequential recurrent steps rather than the $M + N$ sequential steps required for tightly coupled recurrences. Additionally, our approach enables the use of highly optimized one-dimensional RNN implementations.
+
+Various approaches to image generation have succeeded in generating high-resolution, globally coherent images with hundreds of thousands of dimensions (Karras et al., 2017; Reed et al., 2017; Kingma & Dhariwal, 2018). The methods introduced in these works are not directly transferable to waveform generation, as they exploit spatial properties of images which are absent in one-dimensional audio signals. However, these methods are more straightforwardly applicable to two-dimensional representations such as spectrograms. Of particular relevance to our work are approaches which combine autoregressive models with multiscale modelling (van den Oord et al., 2016c; Dahl et al.,
+
+2017; Reed et al., 2017; Menick & Kalchbrenner, 2018). Our work demonstrates that the benefits of a multiscale autoregressive model extend beyond the task of image generation, and can be used to generate high-resolution, globally coherent spectrograms.
+
+# 8 CONCLUSION & FUTURE WORK
+
+We have introduced MelNet, a generative model for spectral representations of audio. MelNet combines a highly expressive autoregressive model with a multiscale modelling scheme to generate high-resolution spectrograms with realistic structure on both local and global scales. In comparison to previous works which model time-domain signals directly, MelNet is particularly well-suited to model long-range temporal dependencies. Experiments show promising results across a diverse set of audio generation tasks.
+
+Furthermore, we believe MelNet provides a foundation for various directions of future work. Two particularly promising directions are text-to-speech synthesis and representation learning:
+
+• Text-to-Speech Synthesis: MelNet utilizes a more flexible probabilistic model than existing end-to-end text-to-speech models, making it well-suited to model expressive, multi-modal speech data.
+
+• Representation Learning: MelNet is able to uncover salient structure from large quantities of unlabelled audio. Large-scale, pre-trained autoregressive models for language modelling have demonstrated significant benefits when fine-tuned for downstream tasks. Likewise, representations learned by MelNet could potentially aid downstream tasks such as speech recognition.
+
+# REFERENCES
+
+Sercan O Arık, Heewoo Jun, and Gregory Diamos. Fast spectrogram inversion using multi-head ¨ convolutional neural networks. IEEE Signal Processing Letters, 26(1):94–98, 2019.
+
+Christopher M Bishop. Mixture density networks. Technical report, Citeseer, 1994.
+
+Xi Chen, Nikhil Mishra, Mostafa Rohaninejad, and Pieter Abbeel. Pixelsnail: An improved autoregressive generative model. arXiv preprint arXiv:1712.09763, 2017.
+
+Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever. Generating long sequences with sparse transformers. arXiv preprint arXiv:1904.10509, 2019.
+
+Joon Son Chung, Arsha Nagrani, and Andrew Zisserman. Voxceleb2: Deep speaker recognition. arXiv preprint arXiv:1806.05622, 2018.
+
+Ryan Dahl, Mohammad Norouzi, and Jonathon Shlens. Pixel recursive super resolution. In Proceedings of the IEEE International Conference on Computer Vision, pp. 5439–5448, 2017.
+
+Remi Decorsi ´ ere, Peter L Søndergaard, Ewen N MacDonald, and Torsten Dau. Inversion of au- \` ditory spectrograms, traditional spectrograms, and other envelope representations. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 23(1):46–56, 2015.
+
+Sander Dieleman, Aaron van den Oord, and Karen Simonyan. The challenge of realistic music generation: modelling raw audio at scale. In Advances in Neural Information Processing Systems, pp. 7999–8009, 2018.
+
+Chris Donahue, Julian McAuley, and Miller Puckette. Synthesizing audio with generative adversarial networks. arXiv preprint arXiv:1802.04208, 2018.
+
+Jesse Engel, Kumar Krishna Agrawal, Shuo Chen, Ishaan Gulrajani, Chris Donahue, and Adam Roberts. Gansynth: Adversarial neural audio synthesis. 2018.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014.
+
+Alex Graves. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850, 2013.
+
+Alex Graves and Jurgen Schmidhuber. Offline handwriting recognition with multidimensional ¨ recurrent neural networks. In Advances in neural information processing systems, pp. 545–552, 2009.
+
+Alex Graves, Santiago Fernandez, and J ´ urgen Schmidhuber. Multi-dimensional recurrent neural ¨ networks. In International conference on artificial neural networks, pp. 549–558. Springer, 2007.
+
+Daniel Griffin and Jae Lim. Signal estimation from modified short-time fourier transform. IEEE Transactions on Acoustics, Speech, and Signal Processing, 32(2):236–243, 1984.
+
+Curtis Hawthorne, Andriy Stasyuk, Adam Roberts, Ian Simon, Cheng-Zhi Anna Huang, Sander Dieleman, Erich Elsen, Jesse Engel, and Douglas Eck. Enabling factorized piano music modeling and generation with the maestro dataset. arXiv preprint arXiv:1810.12247, 2018.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8): 1735–1780, 1997.
+
+Cheng-Zhi Anna Huang, Ashish Vaswani, Jakob Uszkoreit, Noam Shazeer, Curtis Hawthorne, Andrew M Dai, Matthew D Hoffman, and Douglas Eck. Music transformer: Generating music with long-term structure. arXiv preprint arXiv:1809.04281, 2018.
+
+Nal Kalchbrenner, Ivo Danihelka, and Alex Graves. Grid long short-term memory. arXiv preprint arXiv:1507.01526, 2015.
+
+Nal Kalchbrenner, Erich Elsen, Karen Simonyan, Seb Noury, Norman Casagrande, Edward Lockhart, Florian Stimberg, Aaron van den Oord, Sander Dieleman, and Koray Kavukcuoglu. Efficient neural audio synthesis. arXiv preprint arXiv:1802.08435, 2018.
+
+Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. arXiv preprint arXiv:1710.10196, 2017.
+
+Simon King. The blizzard challenge 2011, 2011.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Durk P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. In Advances in Neural Information Processing Systems, pp. 10236–10245, 2018.
+
+Jinyu Li, Abdelrahman Mohamed, Geoffrey Zweig, and Yifan Gong. Exploring multidimensional lstms for large vocabulary asr. In Acoustics, Speech and Signal Processing (ICASSP), 2016 IEEE International Conference on, pp. 4940–4944. IEEE, 2016.
+
+Soroush Mehri, Kundan Kumar, Ishaan Gulrajani, Rithesh Kumar, Shubham Jain, Jose Sotelo, Aaron Courville, and Yoshua Bengio. Samplernn: An unconditional end-to-end neural audio generation model. arXiv preprint arXiv:1612.07837, 2016.
+
+Jacob Menick and Nal Kalchbrenner. Generating high fidelity images with subscale pixel networks and multidimensional upscaling. arXiv preprint arXiv:1812.01608, 2018.
+
+Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Łukasz Kaiser, Noam Shazeer, and Alexander Ku. Image transformer. arXiv preprint arXiv:1802.05751, 2018.
+
+Wei Ping, Kainan Peng, Andrew Gibiansky, Sercan O Arik, Ajay Kannan, Sharan Narang, Jonathan Raiman, and John Miller. Deep voice 3: Scaling text-to-speech with convolutional sequence learning. arXiv preprint arXiv:1710.07654, 2017.
+
+Ryan Prenger, Rafael Valle, and Bryan Catanzaro. Waveglow: A flow-based generative network for speech synthesis. In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3617–3621. IEEE, 2019.
+
+Scott Reed, Aaron van den Oord, Nal Kalchbrenner, Sergio G ¨ omez Colmenarejo, Ziyu Wang, Dan ´ Belov, and Nando de Freitas. Parallel multiscale autoregressive density estimation. arXiv preprint arXiv:1703.03664, 2017.
+
+Tara N Sainath and Bo Li. Modeling time-frequency patterns with lstm vs. convolutional architectures for lvcsr tasks. In INTERSPEECH, pp. 813–817, 2016.
+
+Tim Salimans, Andrej Karpathy, Xi Chen, and Diederik P Kingma. Pixelcnn $^ { + + }$ : Improving the pixelcnn with discretized logistic mixture likelihood and other modifications. arXiv preprint arXiv:1701.05517, 2017.
+
+Jonathan Shen, Ruoming Pang, Ron J Weiss, Mike Schuster, Navdeep Jaitly, Zongheng Yang, Zhifeng Chen, Yu Zhang, Yuxuan Wang, Rj Skerrv-Ryan, et al. Natural tts synthesis by conditioning wavenet on mel spectrogram predictions. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4779–4783. IEEE, 2018.
+
+Jose Sotelo, Soroush Mehri, Kundan Kumar, Joao Felipe Santos, Kyle Kastner, Aaron Courville, and Yoshua Bengio. Char2wav: End-to-end speech synthesis. 2017.
+
+Yaniv Taigman, Lior Wolf, Adam Polyak, and Eliya Nachmani. Voiceloop: Voice fitting and synthesis via a phonological loop. 2018.
+
+Lucas Theis and Matthias Bethge. Generative image modeling using spatial lstms. In Advances in Neural Information Processing Systems, pp. 1927–1935, 2015.
+
+Tijmen Tieleman and Geoffrey Hinton. Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural networks for machine learning, 4(2):26–31, 2012.
+
+Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016a.
+
+Aaron van den Oord, Nal Kalchbrenner, Lasse Espeholt, Oriol Vinyals, Alex Graves, et al. Conditional image generation with pixelcnn decoders. In Advances in Neural Information Processing Systems, pp. 4790–4798, 2016b.
+
+Aaron van den Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. arXiv preprint arXiv:1601.06759, 2016c.
+
+Aaron van den Oord, Oriol Vinyals, et al. Neural discrete representation learning. In Advances in Neural Information Processing Systems, pp. 6306–6315, 2017.
+
+Francesco Visin, Kyle Kastner, Kyunghyun Cho, Matteo Matteucci, Aaron Courville, and Yoshua Bengio. Renet: A recurrent neural network based alternative to convolutional networks. arXiv preprint arXiv:1505.00393, 2015.
+
+Yuxuan Wang, RJ Skerry-Ryan, Daisy Stanton, Yonghui Wu, Ron J Weiss, Navdeep Jaitly, Zongheng Yang, Ying Xiao, Zhifeng Chen, Samy Bengio, et al. Tacotron: Towards end-to-end speech synthesis. arXiv preprint arXiv:1703.10135, 2017.
+
+# A APPENDIX
+
+# A.1 HYPERPARAMETERS & TRAINING DETAILS
+
+All RNNs use LSTM cells (Hochreiter & Schmidhuber, 1997). All models are trained with RMSProp (Tieleman & Hinton, 2012) with a learning rate of $1 0 ^ { - 4 }$ and momentum of 0.9. The initial values for all recurrent states are trainable parameters. A single hyperparameter controls the width of the network—all hidden sizes (RNN state size, residual connections, etc.) are defined by a single value, denoted hidden size in table 2.
+
+Table 2: MelNet hyperparameters.   
+
+<table><tr><td></td><td>Blizzard</td><td>MAESTRO</td><td>VoxCeleb2</td></tr><tr><td>Tiers</td><td>6</td><td>4</td><td>5</td></tr><tr><td>Layers (Initial Tier)</td><td>12</td><td>16</td><td>16</td></tr><tr><td>Layers (Upsampling Tiers)</td><td>5-4-3-2-2</td><td>6-5-4</td><td>6-5-4-3</td></tr><tr><td>Hidden Size</td><td>512</td><td>512</td><td>512</td></tr><tr><td>GMMMixture Components</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Batch Size</td><td>32</td><td>16</td><td>128</td></tr><tr><td>Sample Rate (Hz)</td><td>22.050</td><td>22,050</td><td>16,000</td></tr><tr><td>Max Sample Duration (s)</td><td>10</td><td>6</td><td>6</td></tr><tr><td>Mel Channels</td><td>256</td><td>256</td><td>180</td></tr><tr><td>STFT Hop Size</td><td>256</td><td>256</td><td>180</td></tr><tr><td>STFT Window Size</td><td>6·256</td><td>6·256</td><td>6·180</td></tr></table>
+
+# A.2 WAVENET BASELINE
+
+The human evaluation experiments require samples from a baseline WaveNet model. For the Blizzard and VoxCeleb2 datasets, we use our own reimplementation. Our WaveNet model uses 8-bit $\mu$ -law encoding and models each sample with a discrete distribution. Each model is trained for 150,000 steps. We use the Adam optimizer (Kingma & Ba, 2014) with a learning rate of 0.001 and batch size of 32. Additional hyperparameters are reported in Table 3.
+
+Table 3: WaveNet hyperparameters.   
+
+<table><tr><td></td><td>Blizzard</td><td>VoxCeleb2</td></tr><tr><td>Sample Rate (Hz)</td><td>22,050</td><td>16,000</td></tr><tr><td>Layers</td><td>50</td><td>60</td></tr><tr><td>Kernel Size</td><td>3</td><td>3</td></tr><tr><td>Dilation (at layer i)</td><td>2i mod 10</td><td>2i mod 10</td></tr><tr><td>Residual Channels</td><td>512</td><td>512</td></tr><tr><td>Skip Channels</td><td>512</td><td>512</td></tr><tr><td>Receptive Field (samples)</td><td>10,240</td><td>12,288</td></tr><tr><td>Receptive Field (ms)</td><td>464</td><td>768</td></tr><tr><td>Max Sample Duration (s)</td><td>2</td><td>2</td></tr></table>
+
+We do not use our WaveNet implementation for human evaluation on the MAESTRO dataset. The authors that introduce this dataset provide roughly 2 minutes of audio samples on their website for both unconditional WaveNet and Wave2Midi2Wave models. We generate 50 random 10 second slices from these 2 minutes and directly use them for the human evaluations.

parse/train/r1gIa0NtDH/r1gIa0NtDH_content_list.json

@@ -0,0 +1,1734 @@
+[
+    {
+        "type": "text",
+        "text": "MELNET: A GENERATIVE MODEL FOR AUDIO IN THE FREQUENCY DOMAIN ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            174,
+            400,
+            200
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            238,
+            544,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Capturing high-level structure in audio waveforms is challenging because a single second of audio spans tens of thousands of timesteps. While long-range dependencies are difficult to model directly in the time domain, we show that they can be more tractably modelled in two-dimensional time-frequency representations such as spectrograms. By leveraging this representational advantage, in conjunction with a highly expressive probabilistic model and a multiscale generation procedure, we design a model capable of generating high-fidelity audio samples which capture structure at timescales which time-domain models have yet to achieve. We demonstrate that our model captures longer-range dependencies than time-domain models such as WaveNet across a diverse set of unconditional generation tasks, including single-speaker speech generation, multi-speaker speech generation, and music generation. ",
+        "bbox": [
+            233,
+            273,
+            766,
+            440
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            478,
+            336,
+            493
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Audio waveforms have complex structure at drastically varying timescales, which presents a challenge for generative models. Local structure must be captured to produce high-fidelity audio, while longrange dependencies spanning tens of thousands of timesteps must be captured to generate audio which is globally consistent. Existing generative models of waveforms such as WaveNet (van den Oord et al., 2016a) and SampleRNN (Mehri et al., 2016) are well-adapted to model local dependencies, but as these models typically only backpropagate through a fraction of a second, they are unable to capture high-level structure that emerges on the scale of several seconds. ",
+        "bbox": [
+            174,
+            513,
+            825,
+            609
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce a generative model for audio which captures longer-range dependencies than existing end-to-end models. We primarily achieve this by modelling 2D time-frequency representations such as spectrograms rather than 1D time-domain waveforms (Figure 1). The temporal axis of a spectrogram is orders of magnitude more compact than that of a waveform, meaning dependencies that span tens of thousands of timesteps in waveforms only span hundreds of timesteps in spectrograms. In practice, this enables our spectrogram models to generate unconditional speech and music samples with consistency over multiple seconds whereas time-domain models must be conditioned on intermediate features to capture structure at similar timescales. ",
+        "bbox": [
+            174,
+            617,
+            825,
+            728
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Modelling spectrograms can simplify the task of capturing global structure, but can weaken a model’s ability to capture local characteristics that correlate with audio fidelity. Producing high-fidelity audio has been challenging for existing spectrogram models, which we attribute to the lossy nature of spectrograms and oversmoothing artifacts which result from insufficiently expressive models. To reduce information loss, we model high-resolution spectrograms which have the same dimensionality as their corresponding time-domain signals. To limit oversmoothing, we use a highly expressive autoregressive model which factorizes the distribution over both the time and frequency dimensions. ",
+        "bbox": [
+            174,
+            736,
+            825,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Modelling both fine-grained details and high-level structure in high-dimensional distributions is known to be challenging for autoregressive models. To capture both local and global structure in spectrograms with hundreds of thousands of dimensions, we employ a multiscale approach which generates spectrograms in a coarse-to-fine manner. A low-resolution, subsampled spectrogram that captures high-level structure is generated initially, followed by an iterative upsampling procedure that adds high-resolution details. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/cc6743f8d442787f246b62a0186760052782c67babcb05064f0a46a3806f16d8.jpg",
+        "image_caption": [
+            "Figure 1: Spectrogram and waveform representations of the same 4 second audio signal. The waveform spans nearly 100,000 timesteps whereas the temporal axis of the spectrogram spans roughly 400. Complex structure is nested within the temporal axis of the waveform at various timescales, whereas the spectrogram has structure which is smoothly spread across the time-frequency plane. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            99,
+            812,
+            279
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Combining these representational and modelling techniques yields a highly expressive and broadly applicable generative model of audio. Our contributions are are as follows: ",
+        "bbox": [
+            174,
+            372,
+            825,
+            401
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We introduce MelNet, a generative model for spectrograms which couples a fine-grained autoregressive model and a multiscale generation procedure to jointly capture local and global structure. We show that MelNet is able to model longer-range dependencies than existing time-domain models. Additionally, we include an ablation to demonstrate that multiscale modelling is essential for modelling long-range dependencies.   \nWe demonstrate that MelNet is broadly applicable to a variety of audio generation tasks, including unconditional speech and music generation. Furthermore, MelNet is able to model highly multimodal data such as multi-speaker and multilingual speech. ",
+        "bbox": [
+            215,
+            412,
+            825,
+            549
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            571,
+            339,
+            588
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We briefly present background regarding spectral representations of audio. Audio is represented digitally as a one-dimensional, discrete-time signal $y = ( y _ { 1 } , \\dots , y _ { n } )$ . Existing generative models for audio have predominantly focused on modelling these time-domain signals directly. We instead model spectrograms, which are two-dimensional time-frequency representations which contain information about how the frequency content of an audio signal varies through time. Spectrograms are computed by taking the squared magnitude of the short-time Fourier transform (STFT) of a time-domain signal, i.e. $x = \\| \\mathrm { S T F T } ( y ) \\| ^ { 2 }$ . The value of $x _ { i j }$ (referred to as amplitude or energy) corresponds to the squared magnitude of the $j$ th element of the frequency response at timestep i. Each slice $x _ { i , * }$ is referred to as a frame. We assume a time-major ordering, but following convention, all figures are displayed transposed and with the frequency axis inverted. ",
+        "bbox": [
+            173,
+            603,
+            825,
+            742
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Time-frequency representations such as spectrograms highlight how the tones and pitches within an audio signal vary through time. Such representations are closely aligned with how humans perceive audio. To further align these representations with human perception, we convert the frequency axis to the Mel scale and apply an elementwise logarithmic rescaling of the amplitudes. Roughly speaking, the Mel transformation aligns the frequency axis with human perception of pitch and the logarithmic rescaling aligns the amplitude axis with human perception of loudness. ",
+        "bbox": [
+            173,
+            750,
+            825,
+            833
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Spectrograms are lossy representations of their corresponding time-domain signals. The Mel transformation discards frequency information and the removal of the STFT phase discards temporal information. When recovering a time-domain signal from a spectrogram, this information loss manifests as distortion in the recovered signal. To minimize these artifacts and improve the fidelity of generated audio, we model high-resolution spectrograms. The temporal resolution of a spectrogram can be increased by decreasing the STFT hop size, and the frequency resolution can be increased by increasing the number of Mel channels. Generated spectrograms are converted back to time-domain signals using classical spectrogram inversion algorithms. We experiment with both Griffin-Lim (Griffin & Lim, 1984) and a gradient-based inversion algorithm (Decorsiere et al., 2015), and ultimately \\` use the latter as it generally produced audio with fewer artifacts. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/aa045a4ec7d0df28be1be4bc52a74c1d3b9eb7ed701c13f29cacc84fbac62140.jpg",
+        "image_caption": [
+            "Figure 2: The context $x _ { < i j }$ (grey) for the element $\\boldsymbol { x } _ { i j }$ (black) is encoded using 4 RNNs. Three of these are used in the time-delayed stack to extract features from preceding frames. The fourth is used in the frequency-delayed stack to extract features from all preceding elements within the current frame. Each arrow denotes an individual RNN cell and arrows of the same color use shared parameters. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            99,
+            820,
+            194
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            286,
+            826,
+            343
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PROBABILISTIC MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            362,
+            403,
+            378
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We use an autoregressive model which factorizes the joint distribution over a spectrogram $x$ as a product of conditional distributions. Given an ordering of the dimensions of $x$ , we define the context $x _ { < i j }$ as the elements of $x$ that precede $x _ { i j }$ . We default to a row-major ordering which proceeds through each frame $x _ { i , * }$ from low to high frequency, before progressing to the next frame. The joint density is factorized as ",
+        "bbox": [
+            173,
+            393,
+            825,
+            463
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ade37f5733b0e4fbb78b0cff8381d9a74ce0143bcf20dabf43c7667cdf6464d.jpg",
+        "text": "$$\np ( \\boldsymbol x ) = \\prod _ { i } \\prod _ { j } p ( x _ { i j } \\mid \\boldsymbol x _ { < i j } ; \\theta _ { i j } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            462,
+            611,
+            497
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta _ { i j }$ parameterizes a univariate density over $x _ { i j }$ . We model each factor distribution as a Gaussian mixture model with $K$ components. Thus, $\\theta _ { i j }$ consists of $3 K$ parameters corresponding to means $\\{ \\mu _ { i j k } \\} _ { k = 1 } ^ { K }$ , standard deviations $\\{ \\sigma _ { i j k } \\} _ { k = 1 } ^ { K }$ , and mixture coefficients $\\{ \\pi _ { i j k } \\} _ { k = 1 } ^ { K }$ . The resulting factor distribution can then be expressed as ",
+        "bbox": [
+            173,
+            498,
+            825,
+            558
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3615a5abcbf2fd53cca746e1a79fd8adc81c3d30b8f0ba41fc6145d9398f36c4.jpg",
+        "text": "$$\np ( x _ { i j } \\mid x _ { < i j } ; \\theta _ { i j } ) = \\sum _ { k = 1 } ^ { K } \\pi _ { i j k } \\mathcal { N } ( x _ { i j } ; \\mu _ { i j k } , \\sigma _ { i j k } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            561,
+            663,
+            606
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Following the work on Mixture Density Networks (Bishop, 1994) and their application to autoregressive models (Graves, 2013), $\\theta _ { i j }$ is modelled as the output of a neural network and computed as a function of the context $x _ { < i j }$ . Precisely, for some network $f$ with parameters $\\psi$ , we have $\\theta _ { i j } = f ( \\boldsymbol x _ { < i j } ; \\ \\psi )$ . A maximum-likelihood estimate for the network parameters is computed by minimizing the negative log-likelihood via gradient descent. ",
+        "bbox": [
+            173,
+            609,
+            825,
+            681
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To ensure that the network output parameterizes a valid Gaussian mixture model, the network first computes unconstrained parameters $\\{ \\hat { \\mu } _ { i j k } , \\hat { \\sigma } _ { i j k } , \\hat { \\pi } _ { i j k } \\} _ { k = 1 } ^ { K }$ as a vector $\\hat { \\theta } _ { i j } ~ \\in ~ \\mathbb { R } ^ { 3 K }$ , and enforces constraints on $\\theta _ { i j }$ by applying the following transformations: ",
+        "bbox": [
+            174,
+            686,
+            825,
+            732
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/84da46e7b936dc4250a4249d591bd55288527329505689fbfecff186d444ec30.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mu _ { i j k } = \\hat { \\mu } _ { i j k } } \\\\ & { \\sigma _ { i j k } = \\exp ( \\hat { \\sigma } _ { i j k } ) } \\\\ & { \\pi _ { i j k } = \\cfrac { \\exp \\left( \\hat { \\pi } _ { i j k } \\right) } { \\sum _ { k = 1 } ^ { K } \\exp \\left( \\hat { \\pi } _ { i j k } \\right) } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            410,
+            737,
+            588,
+            813
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "These transformations ensure the standard deviations $\\sigma _ { i j k }$ are positive and the mixture coefficients $\\pi _ { i j k }$ sum to one. ",
+        "bbox": [
+            171,
+            815,
+            825,
+            844
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 NETWORK ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            864,
+            431,
+            881
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To model the distribution in an autoregressive manner, we design a network which computes the distribution over $x _ { i j }$ as a function of the context $x _ { < i j }$ . The network architecture draws inspiration from existing autoregressive models for images (Theis & Bethge, 2015; van den Oord et al., 2016c;b; Chen et al., 2017; Salimans et al., 2017; Parmar et al., 2018; Child et al., 2019). In the same way that these models estimate a distribution pixel-by-pixel over the spatial dimensions of an image, our model estimates a distribution element-by-element over the time and frequency dimensions of a spectrogram. A noteworthy distinction is that spectrograms are not invariant to translation along the frequency axis, making 2D convolution less desirable than other 2D network primitives which do not assume invariance. Utilizing multidimensional recurrence instead of 2D convolution has been shown to be beneficial when modelling spectrograms in discriminative settings (Li et al., 2016; Sainath & Li, 2016), which motivates our use of an entirely recurrent architecture. ",
+        "bbox": [
+            173,
+            895,
+            825,
+            925
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            229
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similar to Gated PixelCNN (van den Oord et al., 2016b), the network has multiple stacks of computation. These stacks extract features from different segments of the input to collectively summarize the full context x<ij : ",
+        "bbox": [
+            176,
+            236,
+            825,
+            279
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• The time-delayed stack computes features which aggregate information from all previous frames x<i,∗. • The frequency-delayed stack utilizes all preceding elements within a frame, $x _ { i , < j }$ , as well as the outputs of the time-delayed stack, to summarize the full context $x _ { < i j }$ . ",
+        "bbox": [
+            217,
+            289,
+            825,
+            351
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The stacks are connected at each layer of the network, meaning that the features generated by layer $l$ of the time-delayed stack are used as input to layer $l$ of the frequency-delayed stack. To facilitate the training of deeper networks, both stacks use residual connections (He et al., 2016). The outputs of the final layer of the frequency-delayed stack are used to compute the unconstrained parameters $\\hat { \\theta }$ . ",
+        "bbox": [
+            178,
+            362,
+            825,
+            420
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 TIME-DELAYED STACK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            436,
+            377,
+            452
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The time-delayed stack utilizes multiple layers of multidimensional RNNs to extract features from $x _ { < i , * }$ , the two-dimensional region consisting of all frames preceding $x _ { i j }$ . Each multidimensional RNN is composed of three one-dimensional RNNs: one which runs forwards along the frequency axis, one which runs backwards along the frequency axis, and one which runs forwards along the time axis. Each RNN runs along each slice of a given axis, as shown in Figure 2. The output of each layer of the time-delayed stack is the concatenation of the three RNN hidden states. ",
+        "bbox": [
+            173,
+            463,
+            825,
+            546
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We denote the function computed at layer $l$ of the time-delayed stack (three RNNs followed by concatenation) as $\\mathcal { F } _ { l } ^ { t }$ . At each layer, the time-delayed stack uses the feature map from the previous layer, $h ^ { t } [ l - 1 ]$ , to compute the subsequent feature map $\\mathcal { F } _ { l } ^ { t } \\big ( h ^ { t } [ l - 1 ] \\big )$ which consists of the three concatenated RNN hidden states. When using residual connections, the computation of $h ^ { t } [ l ]$ from $h ^ { t } [ l - 1 ]$ becomes ",
+        "bbox": [
+            174,
+            553,
+            825,
+            613
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/365b42ab56d0634a8ff2577ac61165eb57062859238f19b37b467a45d749b8f7.jpg",
+        "text": "$$\nh _ { i j } ^ { t } [ l ] = W _ { l } ^ { t } \\mathcal { F } _ { l } ^ { t } \\big ( h ^ { t } [ l - 1 ] \\big ) _ { i j } + h _ { i j } ^ { t } [ l - 1 ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            619,
+            635,
+            641
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To ensure the output $h _ { i j } ^ { t } [ l ]$ is only a function of frames which lie in the context $x _ { < i j }$ , the inputs to the time-delayed stack are shifted backwards one step in time: $h _ { i j } ^ { t } [ 0 ] = W _ { 0 } ^ { t } x _ { i - 1 , j }$ . ",
+        "bbox": [
+            173,
+            647,
+            825,
+            679
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 FREQUENCY-DELAYED STACK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            694,
+            424,
+            709
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The frequency-delayed stack is a one-dimensional RNN which runs forward along the frequency axis. Much like existing one-dimensional autoregressive models (language models, waveform models, etc.), the frequency-delayed stack operates on a one-dimensional sequence (a single frame) and estimates the distribution for each element conditioned on all preceding elements. The primary difference is that it is also conditioned upon the outputs of the time-delayed stack, allowing it to use the full two-dimensional context $x _ { < i j }$ . ",
+        "bbox": [
+            173,
+            719,
+            826,
+            806
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We denote the function computed by the frequency-delayed stack as $\\mathcal { F } _ { l } ^ { f }$ . At each layer, the frequencydelayed stack takes two inputs: the the previous-layer outputs of the frequency-delayed stack, $h _ { i j } ^ { f } [ l - 1 ]$ , and the current-layer outputs of the time-delayed stack $h _ { i j } ^ { t } [ l ]$ . These inputs are summed and used as input to a one-dimensional RNN to produce the output feature map $\\mathcal { F } _ { l } ^ { f } \\left( h ^ { f } [ l - 1 ] , ~ h ^ { t } [ l ] \\right)$ which consists of the RNN hidden state: ",
+        "bbox": [
+            173,
+            813,
+            826,
+            895
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b542fad95e82d85f7c2a652540983c42db287c970da51dcc6bc017d67d71c053.jpg",
+        "text": "$$\nh _ { i j } ^ { f } [ l ] = W _ { l } ^ { f } \\mathcal { F } _ { l } ^ { f } \\big ( h ^ { f } [ l - 1 ] , ~ h ^ { t } [ l ] \\big ) _ { i j } + h _ { i j } ^ { f } [ l - 1 ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            898,
+            661,
+            922
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/0d7e0c8d009dd29dc0c0c3cb3679f00ab978dd75546732c36aaa4963fe08fbe5.jpg",
+        "image_caption": [
+            "Figure 3: Computation graph for a single layer of the network. $\\mathcal { F } _ { l } ^ { t }$ and $\\mathcal { F } _ { l } ^ { f }$ are the functions computed by the time-delayed stack and frequency-delayed stack, respectively. The outputs of these functions are projected (by the matrices $\\it { W } _ { l } ^ { t }$ and $\\boldsymbol { W } _ { l } ^ { f }$ ) and summed with the layer inputs to form residual blocks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            99,
+            503,
+            358
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To ensure that $h _ { i j } ^ { f } [ l ]$ is computed using only elements in the context $x _ { < i j }$ , the inputs to the frequencydelayed stack are shifted backwards one step along the frequency axis: $h _ { i j } ^ { f } [ 0 ] = W _ { 0 } ^ { f } x _ { i , j - 1 }$ . At the final layer, layer $L$ , a linear map is applied to the output of the frequency-delayed stack to produce the unconstrained Gaussian mixture model parameters, i.e. $\\hat { \\theta } _ { i j } = \\bar { W } _ { \\theta } h _ { i j } ^ { f } [ L ]$ . ",
+        "bbox": [
+            173,
+            385,
+            826,
+            452
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 CONDITIONING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            472,
+            323,
+            486
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To incorporate conditioning information into the model, conditioning features $z$ are simply projected onto the input layer along with the inputs $x$ : ",
+        "bbox": [
+            173,
+            500,
+            826,
+            529
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4bef8140c1fb4852715d070f8534f1850302346080bea40aee70b941a295dc54.jpg",
+        "text": "$$\n\\begin{array} { r l } & { h _ { i j } ^ { t } [ 0 ] = W _ { 0 } ^ { t } x _ { i - 1 , j } + W _ { z } ^ { t } z _ { i j } } \\\\ & { h _ { i j } ^ { f } [ 0 ] = W _ { 0 } ^ { f } x _ { i , j - 1 } + W _ { z } ^ { f } z _ { i j } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            537,
+            599,
+            583
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Reshaping, upsampling, and broadcasting can be used as necessary to ensure the conditioning features have the same time and frequency shape as the input spectrogram, e.g. a one-hot vector representation for speaker ID would first be broadcast along both the time and frequency axes. ",
+        "bbox": [
+            174,
+            590,
+            825,
+            633
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 MULTISCALE MODELLING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            656,
+            426,
+            674
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To improve audio fidelity, we generate high-resolution spectrograms which have the same dimensionality as their corresponding time-domain representations. Under this regime, a single training example has several hundreds of thousands of dimensions. Capturing global structure in such high-dimensional distributions is challenging for autoregressive models, which are biased towards capturing local dependencies. To counteract this, we utilize a multiscale approach which effectively permutes the autoregressive ordering so that a spectrogram is generated in a coarse-to-fine order. ",
+        "bbox": [
+            173,
+            690,
+            825,
+            776
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The elements of a spectrogram $x$ are partitioned into $G$ tiers $x ^ { 1 } , \\ldots , x ^ { G }$ , such that each successive tier contains higher-resolution information. We define $x ^ { < g }$ as the union of all tiers which precede $x ^ { g }$ , i.e. $x ^ { < g } = ( x ^ { \\top } , \\ldots , x ^ { g - 1 } )$ . The distribution is factorized over tiers: ",
+        "bbox": [
+            174,
+            781,
+            825,
+            825
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bdb757d9d2bb612890a7c2141910b23fcee40c6084887f768079e76a6df93fdb.jpg",
+        "text": "$$\np ( x ; \\psi ) = \\prod _ { g } p ( x ^ { g } \\mid x ^ { < g } ; \\psi ^ { g } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            835,
+            606,
+            871
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "and the distribution of each tier is further factorized element-by-element as described in Section 3. We explicitly include the parameterization by $\\boldsymbol { \\psi } = ( \\psi ^ { 1 } , \\dots , \\psi ^ { G } )$ to indicate that each tier is modelled by a separate network. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/e5ad1d7dc090959b41614a197b15f6dc0e0ed81fefbd8a20c77ac7d878e9b434.jpg",
+        "image_caption": [
+            "Figure 4: A sampled spectrogram viewed at different stages of the multiscale generation procedure. The initial tier dictates high-level structure and subsequent tiers add fine-grained details. Each upsampling tier doubles the resolution of the spectrogram. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            101,
+            823,
+            215
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/662744988b39d6711b2b2d20475d678c6e982221558b25fe7fa43b3f9f632c7c.jpg",
+        "image_caption": [
+            "Figure 5: Schematic showing how tiers of the multiscale model are interleaved and used to condition the distribution for the subsequent tier. a) The initial tier is generated unconditionally. b) The second tier is generated conditionally given the the initial tier. c) The outputs of tiers 1 and 2 are interleaved along the frequency axis and used to condition the generation of tier 3. d) Tier 3 is interleaved along the time axis with all preceding tiers and used to condition the generation of tier 4. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            280,
+            820,
+            353
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            457,
+            287,
+            470
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "During training, the tiers are generated by recursively partitioning a spectrogram into alternating rows along either the time or frequency axis. We define a function split which partitions an input into even and odd rows along a given axis. The initial step of the recursion applies the split function to a spectrogram $x$ , or equivalently $x ^ { < G + 1 }$ , so that the even-numbered rows are assigned to $x ^ { G }$ and the odd-numbered rows are assigned to $x ^ { < G }$ . Subsequent tiers are defined similarly in a recursive manner: ",
+        "bbox": [
+            173,
+            482,
+            825,
+            553
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1d0f385bbf554b3c9f6a3516cebc2f70c7bcfe5dd651edb257f08a9775bc2499.jpg",
+        "text": "$$\nx ^ { g } , x ^ { < g } = { \\tt s p l i t } ( x ^ { < g + 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            555,
+            593,
+            573
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "At each step of the recursion, we model the distribution $p ( x ^ { g } \\mid x ^ { < g } ; \\psi ^ { g } )$ . The final step of the recursion models the unconditional distribution over the initial tier $p ( x ^ { 1 } ; \\psi ^ { 1 } )$ . ",
+        "bbox": [
+            173,
+            575,
+            821,
+            606
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To model the conditional distribution $p ( x ^ { g } \\mid x ^ { < g } ; \\psi ^ { g } )$ , the network at each tier needs a mechanism to incorporate information from the preceding tiers $\\scriptstyle { \\dot { x } } ^ { < g }$ . To this end, we add a feature extraction network which computes features from $x ^ { < g }$ which are used condition the generation of $x ^ { g }$ . We use a multidimensional RNN consisting of four one-dimensional RNNs which run bidirectionally along slices of both axes of the context $x ^ { < g }$ . A layer of the feature extraction network is similar to a layer of the time-delayed stack, but since the feature extraction network is not causal, we include an RNN which runs backwards along the time axis and do not shift the inputs. The hidden states of the RNNs in the feature extraction network are used to condition the generation of $x ^ { g }$ . As each tier doubles the resolution, the features extracted from $x ^ { < g }$ have the same time and frequency shape as $x ^ { g }$ , allowing the conditioning mechanism described in section 4.3 to be used straightforwardly. ",
+        "bbox": [
+            173,
+            611,
+            825,
+            751
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2 SAMPLING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            767,
+            290,
+            781
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To sample from the multiscale model we iteratively sample a value for $x ^ { g }$ conditioned on $x ^ { < g }$ using the learned distributions defined by the estimated network parameters $\\hat { \\psi } = ( \\hat { \\psi } ^ { 1 } , \\dots , \\hat { \\psi } ^ { G } )$ . The initial tier, $x ^ { 1 }$ , is generated unconditionally by sampling from $p ( x ^ { 1 } ; \\hat { \\psi } ^ { 1 } )$ and subsequent tiers are sampled from $p ( x ^ { g } \\mid x ^ { < g } ; { \\hat { \\psi } } ^ { g } )$ . At each tier, the sampled $x ^ { g }$ is interleaved with the context $x ^ { < g }$ : ",
+        "bbox": [
+            173,
+            792,
+            825,
+            857
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a9272733affc440a12375a8aec27488a20c5b60c945727f872fc1444b0c4b25.jpg",
+        "text": "$$\n\\begin{array} { r } { x ^ { < g + 1 } = \\mathrm { i n t e r l e a v e } ( x ^ { g } , x ^ { < g } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            859,
+            619,
+            877
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The interleave function is simply the inverse of the split function. Sampling terminates once a full spectrogram, $x ^ { < G + 1 }$ , has been generated. A spectrogram generated by a multiscale model is shown in Figure 4 and the sampling procedure is visualized schematically in Figure 5. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            326,
+            117
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the MelNet is broadly applicable as a generative model for audio, we train the model on a diverse set of audio generation tasks (single-speaker speech generation, multi-speaker speech generation, and music generation) using three publicly available datasets. Generated audio samples for each task are available on the accompanying web page https://audio-samples.github.io. We include samples generated using the priming and biasing procedures described by Graves (2013). Biasing lowers the temperature of the predictive distribution and priming seeds the model state with a given sequence of audio prior to sampling. Hyperparameters for all experiments are available in Appendix A. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            231
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Speech and music have rich hierarchies of latent structure. Speech has complex linguistic structure (phonemes, words, syntax, semantics, etc.) and music has highly compositional musical structure (notes, chords, melody and rhythm, etc.). The presence of these latent structures in generated samples can be used as a proxy for how well a generative model has learned dependencies at various timescales. As such, a qualitative analysis of unconditional samples is an insightful method of evaluating generative models of audio. To facilitate such a qualitative evaluation, we train MelNet on each of the three unconditional generation tasks and include samples on the accompanying web page. For completeness, we briefly provide some of our own qualitative observations regarding the generated samples (Sections 6.1, 6.2, and 6.3). In addition to qualitative analysis, we conduct a human evaluation experiment to quantitatively compare how well WaveNet and MelNet capture high-level structure (Section 6.4). Lastly, we ablate the impact of the multiscale generation procedure on MelNet’s ability model long-range dependencies (Section 6.5). ",
+        "bbox": [
+            174,
+            238,
+            825,
+            405
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.1 SINGLE-SPEAKER SPEECH",
+        "text_level": 1,
+        "bbox": [
+            176,
+            421,
+            398,
+            435
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To test MelNet’s ability to model a single speaker in a controlled environment, we utilize the Blizzard 2013 dataset (King, 2011), which consists of audiobook narration performed in a highly animated manner by a professional speaker. We find that MelNet frequently generates samples that contain coherent words and phrases. Even when the model generates incoherent speech, the intonation, prosody, and speaking style remain consistent throughout the duration of the sample. Furthermore, the model learns to produce speech using a variety of character voices and learns to generate samples which contain elements of narration and dialogue. Biased samples tend to contain longer strings of comprehensible words but are read in a less expressive fashion. When primed with a real sequence of audio, MelNet is able to continue sampling speech which has consistent speaking style and intonation. ",
+        "bbox": [
+            174,
+            446,
+            825,
+            573
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.2 MULTI-SPEAKER SPEECH ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            590,
+            392,
+            604
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Audiobook data is recorded in a highly controlled environment. To demonstrate MelNet’s capacity to model distributions with significantly more variation, we utilize the VoxCeleb2 dataset (Chung et al., 2018). The VoxCeleb2 dataset consists of over 2,000 hours of speech data captured with real world noise including laughter, cross-talk, channel effects, music and other sounds. The dataset is also multilingual, with speech from speakers of 145 different nationalities, covering a wide range of accents, ages, ethnicities and languages. When trained on the VoxCeleb2 dataset, we find that MelNet is able to generate unconditional samples with significant variation in both speaker characteristics (accent, language, prosody, speaking style) as well as acoustic conditions (background noise and recording quality). While the generated speech is often not comprehensible, samples can often be identified as belonging to a specific language, indicating that the model has learned distinct modalities for different languages. Furthermore, it is difficult to distinguish real and fake samples which are spoken in foreign languages. For foreign languages, semantic structures are not understood by the listener and cannot be used to discriminate between real and fake. Consequently, the listener must rely largely on phonetic structure, which MelNet is able to realistically model. ",
+        "bbox": [
+            174,
+            616,
+            825,
+            810
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.3 MUSIC ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            828,
+            263,
+            842
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To show that MelNet can model audio modalities other than speech, we apply the model to the task of unconditional music generation. We utilize the MAESTRO dataset (Hawthorne et al., 2018), which consists of over 172 hours of solo piano performances. The samples demonstrate that MelNet learns musical structures such as melody and harmony. Furthermore, generated samples often maintain consistent tempo and contain interesting variation in volume, timbre, and rhythm. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/93aa94a8ccd106583e10508b30978593c69e8b642eb867c4d86c10ee14834bbe.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(a) Comparison between MelNet and WaveNet. Both models are trained in an entirely unsupervised manner. "
+        ],
+        "table_body": "<table><tr><td colspan=\"2\">WaveNet</td><td>MelNet</td></tr><tr><td>Blizzard</td><td>0.0%</td><td>100.0%</td></tr><tr><td>VoxCeleb2</td><td>0.0%</td><td>100.0%</td></tr><tr><td>MAESTRO</td><td>4.2%</td><td>95.8%</td></tr></table>",
+        "bbox": [
+            178,
+            111,
+            431,
+            183
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/f01e06e811ff5dbe463ed2a3ee0a1990086d401e187bba96644f23ccfd707cb5.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(b) Comparison between MelNet and Wave2Midi2Wave. Wave2Midi2Wave is a two-stage model consisting of a Music Transformer trained on labelled MIDI followed by a conditional WaveNet model. The MelNet model, on the other hand, is trained without any intermediate supervision. "
+        ],
+        "table_body": "<table><tr><td colspan=\"2\">Wave2Midi2Wave</td><td>MelNet</td></tr><tr><td>MAESTRO</td><td>37.7%</td><td>62.3 %</td></tr></table>",
+        "bbox": [
+            500,
+            111,
+            808,
+            155
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Selection rates of human evaluators when asked to identify which model generates samples with longer-term structure. Results show that MelNet captures long-range structure better than WaveNet. Furthermore, MelNet outperforms a two-stage model which conditions WaveNet on generated MIDI. ",
+        "bbox": [
+            173,
+            247,
+            825,
+            304
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.4 HUMAN EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            343,
+            364,
+            358
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Making quantitative comparisons with existing generative models such as WaveNet is difficult for various reasons and previous works have ultimately relied on largely empirical evaluations by the reader (Dieleman et al., 2018). To allow the reader to make these judgements for themselves, we provide samples from both WaveNet and MelNet for each of the tasks described in the previous sections. Furthermore, in an effort to provide quantitative metrics to support the claim that MelNet generates samples with improved long-range structure in comparison to WaveNet, we conduct a human experiment whereby participants are presented anonymized samples from both models and asked to select which sample exhibits longer-term structure. We resort to such evaluations since standard metrics for evaluation of generative models such as density estimates cannot be used to compare WaveNet and MelNet as that these models operate on different representations. ",
+        "bbox": [
+            174,
+            375,
+            825,
+            513
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The methodology for this experiment is as follows. For each of the three unconditional audio generation tasks, we generated 50 samples from WaveNet and 50 samples from MelNet. Participants were shown an anonymized, randomly-drawn sample from each model and instructed to “select the sample which has more coherent long-term structure.” We collected 50 evaluations for each task. Results, shown in Table 1a, show that evaluators overwhelmingly agreed that samples generated by MelNet had more coherent long-range structure than samples from WaveNet across all tasks. ",
+        "bbox": [
+            174,
+            521,
+            825,
+            604
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In addition to comparing MelNet to an unconditional WaveNet model for music generation, we also compare to a two-stage Wave2Midi2Wave model (Hawthorne et al., 2018) which conditions WaveNet on MIDI generated by a separately-trained Music Transformer (Huang et al., 2018). The two-stage Wave2Midi2Wave model has the advantage of directly modelling labelled musical notes which distill much of the salient, high-level structure in music into a compact symbolic representation. Despite this, as shown by the results in Table 1b, the two-stage model does not capture long-range structure as well as a MelNet model that is trained without access to any intermediate representations. ",
+        "bbox": [
+            174,
+            611,
+            825,
+            709
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.5 ABLATION: MULTISCALE MODELLING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            739,
+            482,
+            753
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To isolate the impact of multiscale modelling procedure described in Section 5, we train models with varying numbers of tiers and evaluate the long-term coherence of their respective samples. As noted before, long-term coherence is difficult to quantify and we provide samples on the accompanying web page so that the reader can make their own judgements. We believe the samples clearly demonstrate that increasing the number of tiers results in samples with more coherent high-level structure. We note that our experiment varies the number of tiers from two to five. Training a single-tier model on full-resolution spectrograms was prohibitively expensive in terms of memory consumption. This highlights another benefit of multiscale modelling—large, deep networks can be allocated to learning complex distributional structure in the initial tiers while shallower networks can be used for modelling the relatively simple, low-entropy distributions in the upsampling tiers. This allows multiscale models to effectively allocate network capacity in proportion to the complexity of the modelling task. ",
+        "bbox": [
+            174,
+            770,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            344,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The predominant line of research regarding generative models for audio has been directed towards modelling time-domain waveforms with autoregressive models (van den Oord et al., 2016a; Mehri et al., 2016; Kalchbrenner et al., 2018). WaveNet is a competitive baseline for audio generation, and as such, is used for comparison in many of our experiments. However, we note that the contribution of our work is in many ways complementary to that of WaveNet. MelNet is more proficient at capturing high-level structure, whereas WaveNet is capable of producing higher-fidelity audio. Several works have demonstrated that time-domain models can be used to invert spectral representations to highfidelity audio (Shen et al., 2018; Prenger et al., 2019; Arık et al., 2019), suggesting that MelNet could be used in concert with time-domain models such as WaveNet. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            258
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Dieleman et al. (2018) and van den Oord et al. (2017) capture long-range dependencies in waveforms by utilizing a hierarchy of autoencoders. This approach requires multiple stages of models which must be trained sequentially, whereas the multiscale approach in this work can be parallelized over tiers. Additionally, these approaches do not directly optimize the data likelihood, nor do they admit tractable marginalization over the latent codes. We also note that the modelling techniques devised in these works can be broadly applied to autoregressive models such as ours, making their contributions largely complementary to ours. ",
+        "bbox": [
+            174,
+            267,
+            825,
+            366
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Recent works have used generative adversarial networks (GANs) (Goodfellow et al., 2014) to model both waveforms and spectral representations (Donahue et al., 2018; Engel et al., 2018). As with image generation, it remains unclear whether GANs capture all modes of the data distribution. Furthermore, these approaches are restricted to generating fixed-duration segments of audio, which precludes their usage in many audio generation tasks. ",
+        "bbox": [
+            174,
+            375,
+            825,
+            444
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Generating spectral representations is common practice for end-to-end text-to-speech models (Ping et al., 2017; Sotelo et al., 2017; Wang et al., 2017; Taigman et al., 2018). However, these models use probabilistic models which are much less expressive than the fine-grained autoregressive model used by MelNet. Consequently, these models are unsuitable for modelling high-entropy, multimodal distributions such as those involved in tasks like unconditional music generation. ",
+        "bbox": [
+            174,
+            453,
+            825,
+            522
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The network architecture used for MelNet is heavily influenced by recent advancements in deep autoregressive models for images. Theis & Bethge (2015) introduced an LSTM architecture for autoregressive modelling of 2D images and van den Oord et al. (2016c) introduced PixelRNN and PixelCNN and scaled up the models to handle the modelling of natural images. Subsequent works in autoregressive image modelling have steadily improved state-of-the-art for image density estimation (van den Oord et al., 2016b; Salimans et al., 2017; Parmar et al., 2018; Chen et al., 2017; Child et al., 2019). We draw inspiration from many of these models, and ultimately design a recurrent architecture of our own which is suitable for modelling spectrograms rather than images. We note that our choice of architecture is not a fundamental contribution of this work. While we have designed the architecture particularly for modelling spectrograms, we did not experimentally validate whether it outperforms existing architectures and make no such claims to this effect. ",
+        "bbox": [
+            174,
+            530,
+            825,
+            683
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We use a multidimensional recurrence in both the time-delayed stack and the upsampling tiers to extract features from two-dimensional inputs. Our multidimensional recurrence is effectively ‘factorized’ as it independently applies one-dimensional RNNs across each dimension. This approach differs from the tightly coupled multidimensional recurrences used by MDRNNs (Graves et al., 2007; Graves & Schmidhuber, 2009) and GridLSTMs (Kalchbrenner et al., 2015) and more closely resembles the approach taken by ReNet (Visin et al., 2015). Our approach allows for efficient training as we can extract features from an $M \\times N$ grid in $\\operatorname* { m a x } ( M , N )$ sequential recurrent steps rather than the $M + N$ sequential steps required for tightly coupled recurrences. Additionally, our approach enables the use of highly optimized one-dimensional RNN implementations. ",
+        "bbox": [
+            173,
+            691,
+            825,
+            818
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Various approaches to image generation have succeeded in generating high-resolution, globally coherent images with hundreds of thousands of dimensions (Karras et al., 2017; Reed et al., 2017; Kingma & Dhariwal, 2018). The methods introduced in these works are not directly transferable to waveform generation, as they exploit spatial properties of images which are absent in one-dimensional audio signals. However, these methods are more straightforwardly applicable to two-dimensional representations such as spectrograms. Of particular relevance to our work are approaches which combine autoregressive models with multiscale modelling (van den Oord et al., 2016c; Dahl et al., ",
+        "bbox": [
+            174,
+            827,
+            825,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "2017; Reed et al., 2017; Menick & Kalchbrenner, 2018). Our work demonstrates that the benefits of a multiscale autoregressive model extend beyond the task of image generation, and can be used to generate high-resolution, globally coherent spectrograms. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "8 CONCLUSION & FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            165,
+            472,
+            183
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We have introduced MelNet, a generative model for spectral representations of audio. MelNet combines a highly expressive autoregressive model with a multiscale modelling scheme to generate high-resolution spectrograms with realistic structure on both local and global scales. In comparison to previous works which model time-domain signals directly, MelNet is particularly well-suited to model long-range temporal dependencies. Experiments show promising results across a diverse set of audio generation tasks. ",
+        "bbox": [
+            174,
+            196,
+            825,
+            281
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we believe MelNet provides a foundation for various directions of future work. Two particularly promising directions are text-to-speech synthesis and representation learning: ",
+        "bbox": [
+            173,
+            287,
+            823,
+            316
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "• Text-to-Speech Synthesis: MelNet utilizes a more flexible probabilistic model than existing end-to-end text-to-speech models, making it well-suited to model expressive, multi-modal speech data. ",
+        "bbox": [
+            217,
+            329,
+            823,
+            371
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "• Representation Learning: MelNet is able to uncover salient structure from large quantities of unlabelled audio. Large-scale, pre-trained autoregressive models for language modelling have demonstrated significant benefits when fine-tuned for downstream tasks. Likewise, representations learned by MelNet could potentially aid downstream tasks such as speech recognition. ",
+        "bbox": [
+            217,
+            375,
+            825,
+            444
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            465,
+            285,
+            482
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sercan O Arık, Heewoo Jun, and Gregory Diamos. Fast spectrogram inversion using multi-head ¨ convolutional neural networks. IEEE Signal Processing Letters, 26(1):94–98, 2019. ",
+        "bbox": [
+            178,
+            489,
+            823,
+            517
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Christopher M Bishop. Mixture density networks. Technical report, Citeseer, 1994. ",
+        "bbox": [
+            176,
+            526,
+            722,
+            541
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xi Chen, Nikhil Mishra, Mostafa Rohaninejad, and Pieter Abbeel. Pixelsnail: An improved autoregressive generative model. arXiv preprint arXiv:1712.09763, 2017. ",
+        "bbox": [
+            174,
+            551,
+            823,
+            580
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever. Generating long sequences with sparse transformers. arXiv preprint arXiv:1904.10509, 2019. ",
+        "bbox": [
+            173,
+            588,
+            825,
+            617
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Joon Son Chung, Arsha Nagrani, and Andrew Zisserman. Voxceleb2: Deep speaker recognition. arXiv preprint arXiv:1806.05622, 2018. ",
+        "bbox": [
+            171,
+            626,
+            825,
+            655
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ryan Dahl, Mohammad Norouzi, and Jonathon Shlens. Pixel recursive super resolution. In Proceedings of the IEEE International Conference on Computer Vision, pp. 5439–5448, 2017. ",
+        "bbox": [
+            171,
+            664,
+            825,
+            694
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Remi Decorsi ´ ere, Peter L Søndergaard, Ewen N MacDonald, and Torsten Dau. Inversion of au- \\` ditory spectrograms, traditional spectrograms, and other envelope representations. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 23(1):46–56, 2015. ",
+        "bbox": [
+            178,
+            702,
+            823,
+            746
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sander Dieleman, Aaron van den Oord, and Karen Simonyan. The challenge of realistic music generation: modelling raw audio at scale. In Advances in Neural Information Processing Systems, pp. 7999–8009, 2018. ",
+        "bbox": [
+            174,
+            753,
+            825,
+            796
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chris Donahue, Julian McAuley, and Miller Puckette. Synthesizing audio with generative adversarial networks. arXiv preprint arXiv:1802.04208, 2018. ",
+        "bbox": [
+            169,
+            805,
+            825,
+            834
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jesse Engel, Kumar Krishna Agrawal, Shuo Chen, Ishaan Gulrajani, Chris Donahue, and Adam Roberts. Gansynth: Adversarial neural audio synthesis. 2018. ",
+        "bbox": [
+            169,
+            843,
+            823,
+            872
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014. ",
+        "bbox": [
+            174,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Graves. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850, 2013. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Alex Graves and Jurgen Schmidhuber. Offline handwriting recognition with multidimensional ¨ recurrent neural networks. In Advances in neural information processing systems, pp. 545–552, 2009. ",
+        "bbox": [
+            174,
+            141,
+            823,
+            184
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Alex Graves, Santiago Fernandez, and J ´ urgen Schmidhuber. Multi-dimensional recurrent neural ¨ networks. In International conference on artificial neural networks, pp. 549–558. Springer, 2007. ",
+        "bbox": [
+            173,
+            194,
+            823,
+            223
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Daniel Griffin and Jae Lim. Signal estimation from modified short-time fourier transform. IEEE Transactions on Acoustics, Speech, and Signal Processing, 32(2):236–243, 1984. ",
+        "bbox": [
+            176,
+            232,
+            821,
+            262
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Curtis Hawthorne, Andriy Stasyuk, Adam Roberts, Ian Simon, Cheng-Zhi Anna Huang, Sander Dieleman, Erich Elsen, Jesse Engel, and Douglas Eck. Enabling factorized piano music modeling and generation with the maestro dataset. arXiv preprint arXiv:1810.12247, 2018. ",
+        "bbox": [
+            174,
+            271,
+            826,
+            314
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016. ",
+        "bbox": [
+            174,
+            324,
+            826,
+            367
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8): 1735–1780, 1997. ",
+        "bbox": [
+            169,
+            376,
+            825,
+            405
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Cheng-Zhi Anna Huang, Ashish Vaswani, Jakob Uszkoreit, Noam Shazeer, Curtis Hawthorne, Andrew M Dai, Matthew D Hoffman, and Douglas Eck. Music transformer: Generating music with long-term structure. arXiv preprint arXiv:1809.04281, 2018. ",
+        "bbox": [
+            174,
+            415,
+            825,
+            458
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Nal Kalchbrenner, Ivo Danihelka, and Alex Graves. Grid long short-term memory. arXiv preprint arXiv:1507.01526, 2015. ",
+        "bbox": [
+            171,
+            467,
+            825,
+            496
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Nal Kalchbrenner, Erich Elsen, Karen Simonyan, Seb Noury, Norman Casagrande, Edward Lockhart, Florian Stimberg, Aaron van den Oord, Sander Dieleman, and Koray Kavukcuoglu. Efficient neural audio synthesis. arXiv preprint arXiv:1802.08435, 2018. ",
+        "bbox": [
+            178,
+            506,
+            825,
+            549
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. arXiv preprint arXiv:1710.10196, 2017. ",
+        "bbox": [
+            174,
+            559,
+            821,
+            588
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Simon King. The blizzard challenge 2011, 2011. ",
+        "bbox": [
+            173,
+            597,
+            493,
+            612
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. ",
+        "bbox": [
+            174,
+            621,
+            825,
+            651
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Durk P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. In Advances in Neural Information Processing Systems, pp. 10236–10245, 2018. ",
+        "bbox": [
+            171,
+            660,
+            823,
+            690
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jinyu Li, Abdelrahman Mohamed, Geoffrey Zweig, and Yifan Gong. Exploring multidimensional lstms for large vocabulary asr. In Acoustics, Speech and Signal Processing (ICASSP), 2016 IEEE International Conference on, pp. 4940–4944. IEEE, 2016. ",
+        "bbox": [
+            173,
+            699,
+            826,
+            742
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Soroush Mehri, Kundan Kumar, Ishaan Gulrajani, Rithesh Kumar, Shubham Jain, Jose Sotelo, Aaron Courville, and Yoshua Bengio. Samplernn: An unconditional end-to-end neural audio generation model. arXiv preprint arXiv:1612.07837, 2016. ",
+        "bbox": [
+            176,
+            751,
+            823,
+            795
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jacob Menick and Nal Kalchbrenner. Generating high fidelity images with subscale pixel networks and multidimensional upscaling. arXiv preprint arXiv:1812.01608, 2018. ",
+        "bbox": [
+            169,
+            804,
+            823,
+            833
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Łukasz Kaiser, Noam Shazeer, and Alexander Ku. Image transformer. arXiv preprint arXiv:1802.05751, 2018. ",
+        "bbox": [
+            168,
+            843,
+            825,
+            872
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Wei Ping, Kainan Peng, Andrew Gibiansky, Sercan O Arik, Ajay Kannan, Sharan Narang, Jonathan Raiman, and John Miller. Deep voice 3: Scaling text-to-speech with convolutional sequence learning. arXiv preprint arXiv:1710.07654, 2017. ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ryan Prenger, Rafael Valle, and Bryan Catanzaro. Waveglow: A flow-based generative network for speech synthesis. In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3617–3621. IEEE, 2019. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Scott Reed, Aaron van den Oord, Nal Kalchbrenner, Sergio G ¨ omez Colmenarejo, Ziyu Wang, Dan ´ Belov, and Nando de Freitas. Parallel multiscale autoregressive density estimation. arXiv preprint arXiv:1703.03664, 2017. ",
+        "bbox": [
+            176,
+            155,
+            823,
+            196
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tara N Sainath and Bo Li. Modeling time-frequency patterns with lstm vs. convolutional architectures for lvcsr tasks. In INTERSPEECH, pp. 813–817, 2016. ",
+        "bbox": [
+            173,
+            207,
+            821,
+            236
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tim Salimans, Andrej Karpathy, Xi Chen, and Diederik P Kingma. Pixelcnn $^ { + + }$ : Improving the pixelcnn with discretized logistic mixture likelihood and other modifications. arXiv preprint arXiv:1701.05517, 2017. ",
+        "bbox": [
+            173,
+            244,
+            825,
+            286
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jonathan Shen, Ruoming Pang, Ron J Weiss, Mike Schuster, Navdeep Jaitly, Zongheng Yang, Zhifeng Chen, Yu Zhang, Yuxuan Wang, Rj Skerrv-Ryan, et al. Natural tts synthesis by conditioning wavenet on mel spectrogram predictions. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4779–4783. IEEE, 2018. ",
+        "bbox": [
+            173,
+            295,
+            826,
+            353
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jose Sotelo, Soroush Mehri, Kundan Kumar, Joao Felipe Santos, Kyle Kastner, Aaron Courville, and Yoshua Bengio. Char2wav: End-to-end speech synthesis. 2017. ",
+        "bbox": [
+            176,
+            361,
+            823,
+            390
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yaniv Taigman, Lior Wolf, Adam Polyak, and Eliya Nachmani. Voiceloop: Voice fitting and synthesis via a phonological loop. 2018. ",
+        "bbox": [
+            174,
+            398,
+            823,
+            428
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Lucas Theis and Matthias Bethge. Generative image modeling using spatial lstms. In Advances in Neural Information Processing Systems, pp. 1927–1935, 2015. ",
+        "bbox": [
+            173,
+            436,
+            825,
+            465
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tijmen Tieleman and Geoffrey Hinton. Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural networks for machine learning, 4(2):26–31, 2012. ",
+        "bbox": [
+            173,
+            473,
+            826,
+            516
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016a. ",
+        "bbox": [
+            174,
+            526,
+            823,
+            569
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Aaron van den Oord, Nal Kalchbrenner, Lasse Espeholt, Oriol Vinyals, Alex Graves, et al. Conditional image generation with pixelcnn decoders. In Advances in Neural Information Processing Systems, pp. 4790–4798, 2016b. ",
+        "bbox": [
+            174,
+            577,
+            823,
+            619
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Aaron van den Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. arXiv preprint arXiv:1601.06759, 2016c. ",
+        "bbox": [
+            169,
+            628,
+            825,
+            657
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Aaron van den Oord, Oriol Vinyals, et al. Neural discrete representation learning. In Advances in Neural Information Processing Systems, pp. 6306–6315, 2017. ",
+        "bbox": [
+            169,
+            666,
+            823,
+            696
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Francesco Visin, Kyle Kastner, Kyunghyun Cho, Matteo Matteucci, Aaron Courville, and Yoshua Bengio. Renet: A recurrent neural network based alternative to convolutional networks. arXiv preprint arXiv:1505.00393, 2015. ",
+        "bbox": [
+            176,
+            704,
+            823,
+            747
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yuxuan Wang, RJ Skerry-Ryan, Daisy Stanton, Yonghui Wu, Ron J Weiss, Navdeep Jaitly, Zongheng Yang, Ying Xiao, Zhifeng Chen, Samy Bengio, et al. Tacotron: Towards end-to-end speech synthesis. arXiv preprint arXiv:1703.10135, 2017. ",
+        "bbox": [
+            176,
+            756,
+            825,
+            799
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 HYPERPARAMETERS & TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            519,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "All RNNs use LSTM cells (Hochreiter & Schmidhuber, 1997). All models are trained with RMSProp (Tieleman & Hinton, 2012) with a learning rate of $1 0 ^ { - 4 }$ and momentum of 0.9. The initial values for all recurrent states are trainable parameters. A single hyperparameter controls the width of the network—all hidden sizes (RNN state size, residual connections, etc.) are defined by a single value, denoted hidden size in table 2. ",
+        "bbox": [
+            174,
+            159,
+            825,
+            229
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/91b8cdbeaf2bd8981ed7097640a5912af3588c0fef9900d5089700c69397b659.jpg",
+        "table_caption": [
+            "Table 2: MelNet hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Blizzard</td><td>MAESTRO</td><td>VoxCeleb2</td></tr><tr><td>Tiers</td><td>6</td><td>4</td><td>5</td></tr><tr><td>Layers (Initial Tier)</td><td>12</td><td>16</td><td>16</td></tr><tr><td>Layers (Upsampling Tiers)</td><td>5-4-3-2-2</td><td>6-5-4</td><td>6-5-4-3</td></tr><tr><td>Hidden Size</td><td>512</td><td>512</td><td>512</td></tr><tr><td>GMMMixture Components</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Batch Size</td><td>32</td><td>16</td><td>128</td></tr><tr><td>Sample Rate (Hz)</td><td>22.050</td><td>22,050</td><td>16,000</td></tr><tr><td>Max Sample Duration (s)</td><td>10</td><td>6</td><td>6</td></tr><tr><td>Mel Channels</td><td>256</td><td>256</td><td>180</td></tr><tr><td>STFT Hop Size</td><td>256</td><td>256</td><td>180</td></tr><tr><td>STFT Window Size</td><td>6·256</td><td>6·256</td><td>6·180</td></tr></table>",
+        "bbox": [
+            250,
+            282,
+            746,
+            465
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 WAVENET BASELINE",
+        "text_level": 1,
+        "bbox": [
+            176,
+            517,
+            366,
+            532
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The human evaluation experiments require samples from a baseline WaveNet model. For the Blizzard and VoxCeleb2 datasets, we use our own reimplementation. Our WaveNet model uses 8-bit $\\mu$ -law encoding and models each sample with a discrete distribution. Each model is trained for 150,000 steps. We use the Adam optimizer (Kingma & Ba, 2014) with a learning rate of 0.001 and batch size of 32. Additional hyperparameters are reported in Table 3. ",
+        "bbox": [
+            173,
+            544,
+            825,
+            613
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/8cdacf76357d601ce2ae70eb625b50b1e712db5156864291400d3b51be1b540a.jpg",
+        "table_caption": [
+            "Table 3: WaveNet hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Blizzard</td><td>VoxCeleb2</td></tr><tr><td>Sample Rate (Hz)</td><td>22,050</td><td>16,000</td></tr><tr><td>Layers</td><td>50</td><td>60</td></tr><tr><td>Kernel Size</td><td>3</td><td>3</td></tr><tr><td>Dilation (at layer i)</td><td>2i mod 10</td><td>2i mod 10</td></tr><tr><td>Residual Channels</td><td>512</td><td>512</td></tr><tr><td>Skip Channels</td><td>512</td><td>512</td></tr><tr><td>Receptive Field (samples)</td><td>10,240</td><td>12,288</td></tr><tr><td>Receptive Field (ms)</td><td>464</td><td>768</td></tr><tr><td>Max Sample Duration (s)</td><td>2</td><td>2</td></tr></table>",
+        "bbox": [
+            316,
+            666,
+            679,
+            823
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We do not use our WaveNet implementation for human evaluation on the MAESTRO dataset. The authors that introduce this dataset provide roughly 2 minutes of audio samples on their website for both unconditional WaveNet and Wave2Midi2Wave models. We generate 50 random 10 second slices from these 2 minutes and directly use them for the human evaluations. ",
+        "bbox": [
+            173,
+            852,
+            825,
+            907
+        ],
+        "page_idx": 12
+    }
+]